Editing Appendix/Ramblings/PPToriPt2A (section)

====Real Parts====

=====TERM1=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{\ell^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2(n+1)[2^3(n+1)\cos^2\theta -3](1+3xb) 
+2^4(n+1)^2(\sin^2\theta - \cos^2\theta)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta\biggl(\frac{x}{\beta}\biggr) \biggl[
-2^4\cdot 3 (n+1)^2\cos^3\theta + 2^4(n+1)^2\cos^5\theta + 3^2(n+1)(16n +19)\sin^2\theta \cos\theta -2^3\cdot 23 (n+1)^2\sin^2\theta \cos^3\theta
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2^4(n+1)^2\cos^2\theta  -6(n+1)
+2^4(n+1)^2(1 - 2\cos^2\theta)
+3b\beta\biggl(\frac{x}{\beta}\biggr)\biggl[2^4(n+1)^2\cos^2\theta -6(n+1) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta\biggl(\frac{x}{\beta}\biggr) \biggl[
-2^4\cdot 3 (n+1)^2\cos^3\theta + 2^4(n+1)^2\cos^5\theta + 3^2(n+1)(16n +19)\sin^2\theta \cos\theta -2^3\cdot 23 (n+1)^2\sin^2\theta \cos^3\theta
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6(n+1)
+2^4(n+1)^2(1 - \cos^2\theta)
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{2^4\cdot 3 (n+1)^2 [3\cos^2\theta -\cos^4\theta] -18(n+1)[3-\cos^2\theta] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-2^4\cdot 3 (n+1)^2\cos^2\theta + 2^4(n+1)^2\cos^4\theta + 3^2(n+1)(16n +19)(1-\cos^2\theta) -2^3\cdot 23 (n+1)^2 (\cos^2\theta - \cos^4\theta)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6(n+1)
+2^4(n+1)^2(1 - \cos^2\theta)
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ 3^2(n+1)(16n +19)  -2\cdot 3^3(n+1)  +  2^4\cdot 3^2 (n+1)^2 \cos^2\theta + 2\cdot 3^2(n+1)\cos^2\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
-2^4\cdot 3 (n+1)^2\cos^2\theta - 3^2(n+1)(16n +19)\cos^2\theta -2^3\cdot 23 (n+1)^2 \cos^2\theta - 2^4\cdot 3 (n+1)^2 \cos^4\theta+ 2^4(n+1)^2\cos^4\theta 
 + 2^3\cdot 23 (n+1)^2 \cos^4\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6(n+1)
+2^4(n+1)^2(1 - \cos^2\theta)
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ 3^2(n+1)(16n +13) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  + \cos^2\theta\biggl[2^3(n+1)^2(~18 -23 -6~) + 3^2(n+1)(~2-16n-19~)  
\biggr]  + 2^3(n+1)^2\cos^4\theta\biggl[ - 2\cdot 3  + 2 + 23  \biggr]
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6(n+1)
+2^4(n+1)^2(1 - \cos^2\theta)
+\beta\biggl(\frac{x}{\beta}\biggr)(n+1)\cos\theta \biggl\{ 3^2(16n +13) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  - \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{(n+1)}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr] \biggl[1 - 2\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2 \cos^2\theta  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} \biggl[1 - 2\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2 \cos^2\theta  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr] 
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl[ 12 - 2^5(n+1) + 2^5(n+1)\cos^2\theta\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2 \cos^2\theta\biggl[ 3 - 2^3(n+1) + 2^3(n+1)\cos^2\theta\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}   
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ (112n +97) 
- \cos^2\theta\biggl[200n + 209  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{(136n +112) 
- \cos^2\theta\biggl[224n + 233  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}   \, .
</math>
  </td>
</tr>
</table>
</div>

=====TERM2=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{\ell}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-6(n+1) + 2^4(n+1)^2\cos^2\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \beta\biggl(\frac{x}{\beta}\biggr) (n+1)\cos\theta \biggl\{ [ 15 + 2^4(n+1) ] -\cos^2\theta[9 + 2^3\cdot 7 (n+1)] +2^3\cdot 3(n+1)\cos^4\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\beta^2\biggl(\frac{x}{\beta}\biggr)^2 (n+1) \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(n+1)\biggl[ -6 + 2^4(n+1)\cos^2\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \beta\biggl(\frac{x}{\beta}\biggr) (n+1)\cos\theta \biggl\{ [ 31 + 16n ] -\cos^2\theta[65 + 56n] +2^3\cdot 3(n+1)\cos^4\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\beta^2\biggl(\frac{x}{\beta}\biggr)^2 (n+1) \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{(n+1)}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \beta\biggl(\frac{x}{\beta}\biggr) \cos\theta \biggl\{ [ 31 + 16n ] -\cos^2\theta[65 + 56n] +2^3\cdot 3(n+1)\cos^4\theta \biggr\}
\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\beta^2\biggl(\frac{x}{\beta}\biggr)^2 \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]
-\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \beta\biggl(\frac{x}{\beta}\biggr) \cos\theta \biggl\{ [ 31 + 16n ] -\cos^2\theta[65 + 56n] +2^3\cdot 3(n+1)\cos^4\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta^2\biggl(\frac{x}{\beta}\biggr)^2  \biggl\{ [ 31 + 16n ]\cos^2\theta - [65 + 56n] \cos^4\theta +2^3\cdot 3(n+1)\cos^6\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\beta^2\biggl(\frac{x}{\beta}\biggr)^2 \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos\theta \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \beta\biggl(\frac{x}{\beta}\biggr) \cos\theta \biggl\{ [ 31 + 16n -6] -\cos^2\theta[65 + 56n] + 2^4(n+1)\cos^2\theta  +2^3\cdot 3(n+1)\cos^4\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta^2\biggl(\frac{x}{\beta}\biggr)^2  \biggl\{9 - [ 5 +56n ]\cos^2\theta - [106 + 88n] \cos^4\theta +2^5(n+1)\cos^6\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos\theta \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>
</table>
</div>


=====Sum of TERM1 and TERM2=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2}}{(n+1)} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr] +\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta^2\biggl(\frac{x}{\beta}\biggr)^2  \biggl\{9 - [ 5 +56n ]\cos^2\theta - [106 + 88n] \cos^4\theta +2^5(n+1)\cos^6\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{(136n +112) 
- \cos^2\theta\biggl[224n + 233  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos\theta \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}  
</math>
  </td>
</tr>
</table>
</div>


=====TERM3=====

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

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\mathrm{TERM3}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  2^2(n+1)^2 +
m^2(4n+1)\beta^2 - m^2 \beta^2\biggl(\frac{x}{\beta}\biggr)^2 (n+1)^2 \biggl[2^3 \cos^2\theta - 3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1)^2 b\biggl[2^3 \cos^2\theta - 3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM3}}{(n+1)}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-  2^2(n+1) +
m^2\biggl[\frac{(4n+1)}{(n+1)}\biggr] \beta^2 - m^2 \beta^2\biggl(\frac{x}{\beta}\biggr)^2 (n+1) \biggl[2^3 \cos^2\theta - 3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

=====Sum of TERM1 + TERM2 + TERM3=====
Therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}}{(n+1)} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr] +\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr] ~-  2^2(n+1) 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \beta^2\biggl(\frac{x}{\beta}\biggr)^2  \biggl\{9 - [ 5 +56n ]\cos^2\theta - [106 + 88n] \cos^4\theta +2^5(n+1)\cos^6\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{(136n +112) 
- \cos^2\theta\biggl[224n + 233  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ m^2\biggl[\frac{(4n+1)}{(n+1)}\biggr] \beta^2 - m^2 \beta^2\biggl(\frac{x}{\beta}\biggr)^2 (n+1) \biggl[2^3 \cos^2\theta - 3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos\theta \biggl\{9 - 2^2\cdot 3^2(1+2n)\cos^2\theta - [9 + 32(n+1)]\cos^4\theta +2^3(n+1)\cos^6\theta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13) 
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr]  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~12n 
+ \beta\biggl(\frac{x}{\beta}\biggr) \cos\theta \biggl\{ 2^3\cdot 3[ 3 + 4n] -2^5\cdot 5(n+1)\cos^2\theta  +2^7(n+1) \cos^4\theta \biggr\} + \mathcal{O}(\beta^2)
</math>
  </td>
</tr>

</table>
</div>

=====TERM4=====
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ (n+1)[2^3(n+1)\cos^2\theta -3]x(2+3xb)\biggr\} \cdot \biggl[ -x(2+3xb) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ (n+1)\sin\theta \biggl\{
-2^4 (n+1) (\beta\eta)^2 \cos\theta + 3x^3 \sin^2\theta \biggl[3 - 2^3(n+1)\cos^2\theta \biggr]
\biggr\} \cdot \biggl[ 3x\sin^3\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~(n+1)[2^3(n+1)\cos^2\theta -3]x^2(2+3xb)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ 3x^3(n+1)\sin^4\theta \biggl\{
2^4 (n+1) (1+xb) \cos\theta + 3x \sin^2\theta [2^3(n+1)\cos^2\theta -3]
\biggr\}  
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ x^3 \cdot 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta (1+xb)  
~-~x^4\cdot 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta -3] \, .</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-x\biggl\{~x[~18.37695315~] + x^2[~72.5625~] + x^3[~7.59375~]~~\biggr\} = -x[~9.24804688~]\, .
</math>
  </td>
</tr>
</table>
</div>

Or, continuing to develop the analytic power-law expression,

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

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2  (n+1)[2^3(n+1)\cos^2\theta -3] \biggl[4 + 12\beta \biggl( \frac{x}{\beta}\biggr)b + 9 \beta^2\biggl( \frac{x}{\beta}\biggr)^2 b^2  \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta \biggl[ 1+\beta \biggl( \frac{x}{\beta}\biggr)b \biggr]  
~-~\beta^4 \biggl( \frac{x}{\beta}\biggr)^4 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta -3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3] 
-~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^2\cdot 3 (n+1)[2^3(n+1)\cos^2\theta -3] b
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta  
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \mathrm{Re}\biggl[\mathrm{TERM4}\biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3] 
-~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^2\cdot 3 (n+1)[2^3(n+1)\cos^2\theta -3] b
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta  
+~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^4 (n+1)[2^3(n+1)\cos^2\theta -3] \cos\theta
</math> \, .
  </td>
</tr>
</table>
</div>

=====TERM5=====
Now, let's examine the TERM5 expressions.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}\biggl[\mathrm{TERM5}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathrm{Re}\biggl[ \ell^4\biggl(\frac{\nu}{m}\biggr)^2 + 2\ell^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] \cdot \mathrm{Re}[ 2^3(n+1)^2 + 2m^2\Lambda ] 
-\mathrm{Im}\biggl[ \ell^4\biggl(\frac{\nu}{m}\biggr)^2 + 2\ell^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] \cdot \mathrm{Im}[ 2^3(n+1)^2 + 2m^2\Lambda ] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<b><font color="red" size="+1">Case B:</font></b>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \ell^4\biggl[1-\frac{3\beta^2}{2(n+1)}\biggr] + 2\ell^2\biggl(-1\biggr)+ 1 \biggr\} \cdot \biggl\{ 2^3(n+1)^2 
+ 2m^2\biggl[ ~- (4n+1)\beta^2 + (n+1)^2(2^3 \cos^2\theta - 3) x^2(1+xb)\biggr]  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\biggl\{ \ell^4(-1)\biggl[\frac{2\cdot 3\beta^2}{(n+1)}\biggr]^{1/2} 
+ 2\ell^2\biggl[ \frac{3\beta^2}{2(n+1)}\biggr]^{1/2} \biggr\} \cdot  2m^2\beta [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot x(1+xb)^{1/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{1 - 2\ell^2 + \ell^4-\frac{3\beta^2\ell^4}{2(n+1)}  \biggr\} \cdot \biggl\{ \biggl[ 2^3(n+1)^2 
- 2m^2(4n+1)\beta^2\biggr] + x^2\cdot 2m^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~x\beta^2 \cdot m^2[\ell^2 - \ell^4 ] \cdot   [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta (1+xb)^{1/2} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{3\beta^2}{2(n+1)} \biggl[  1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 6\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
 + \mathcal{O}(\beta^3)  \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times
\biggl\{ \biggl[ 2^3(n+1)^2 
- 2m^2(4n+1)\beta^2\biggr] + \beta^2 \biggl( \frac{x}{\beta}\biggr)^2\cdot 2m^2(n+1)^2(2^3 \cos^2\theta - 3)  
+ \beta^3 \biggl( \frac{x}{\beta}\biggr)^3\cdot 2m^2(n+1)^2(2^3 \cos^2\theta - 3) b \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^3\biggl(\frac{x}{\beta}\biggr) \cdot m^2 [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta \biggl[ \beta^0(1-1) + 2\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta - 5\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{\beta^0(1-2+1)  
+ (4-4)\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + (6-2)\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
- 4\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta + \beta^4\biggl(\frac{x}{\beta}\biggr)^4\cos^4\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{3\beta^2}{2(n+1)} \biggl[  1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 6\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
 + \mathcal{O}(\beta^3)  \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times \biggl\{ 2^3(n+1)^2 
+ 2m^2\beta^2\biggl[- (4n+1) + \biggl( \frac{x}{\beta}\biggr)^2 (n+1)^2(2^3 \cos^2\theta - 3)  \biggr]
+ \beta^3 \biggl( \frac{x}{\beta}\biggr)^3\cdot 2m^2(n+1)^2(2^3 \cos^2\theta - 3) b \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^3\biggl(\frac{x}{\beta}\biggr) \cdot m^2 [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta \biggl[ \beta^0(1-1) + 2\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta - 5\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl\{ 4\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
-\frac{3\beta^2}{2(n+1)} - 4\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta  
+\frac{2\cdot 3\beta^3}{(n+1)} \biggl(\frac{x}{\beta}\biggr)\cos\theta  + \mathcal{O}(\beta^4) \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~\times \biggl\{ 2^3(n+1)^2 
+ 2m^2\beta^2\biggl[- (4n+1) + \biggl( \frac{x}{\beta}\biggr)^2 (n+1)^2(2^3 \cos^2\theta - 3)  \biggr]
+ \beta^3 \biggl( \frac{x}{\beta}\biggr)^3\cdot 2m^2(n+1)^2(2^3 \cos^2\theta - 3) b \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^4\biggl(\frac{x}{\beta}\biggr) \cdot m^2 [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta \biggl[ 2 \biggl(\frac{x}{\beta}\biggr)\cos\theta - 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~2^3(n+1)^2
\biggl\{ 4\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
-\frac{3\beta^2}{2(n+1)} - 4\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta  
+\frac{2\cdot 3\beta^3}{(n+1)} \biggl(\frac{x}{\beta}\biggr)\cos\theta  + \mathcal{O}(\beta^4) \biggr\} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Sum of TERM4 and TERM5=====

When added together, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Re}[\mathrm{TERM4} + \mathrm{TERM5}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl(\frac{x}{\beta}\biggr)^2 \ell^4\cdot 2^2 (n+1)[2^3(n+1)\cos^2\theta -3 ]\biggl(1+\frac{3xb}{2}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \beta^3 \biggl(\frac{x}{\beta}\biggr)^3\ell^4\cdot 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta (1+xb)  
~-~\beta^4\biggl(\frac{x}{\beta}\biggr)^4 \ell^4\cdot 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta-3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~\biggl\{1 - 2\ell^2 + \ell^4 \biggr\} \cdot \biggl\{ 2^3(n+1)^2 + 2m^2\beta^2\biggr[
- (4n+1) + \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\frac{3\beta^2\ell^4}{2(n+1)} \biggl\{ 2^3(n+1)^2 + 2m^2\beta^2\biggr[
- (4n+1) + \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^3\biggl(\frac{x}{\beta}\biggr) \cdot m^2[\ell^2 - \ell^4 ] \cdot   [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta (1+xb)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2\ell^2 + \ell^4 \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2\ell^2 + \ell^4 ] \cdot \biggr[
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^2\ell^4 2^2\cdot 3 (n+1) 
+ \beta^2 \biggl(\frac{x}{\beta}\biggr)^2 \ell^4\cdot 2^2 (n+1)[3 - 2^3(n+1)\cos^2\theta ]\biggl(1+\frac{3xb}{2}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\cancelto{0}{\beta^3}\biggl(\frac{x}{\beta}\biggr) \cdot m^2[\ell^2 - \ell^4 ] \cdot   [  2^{10}\cdot 3^2(n+1)^2 ]^{1/2} \cos\theta (1+xb)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \cancelto{0}{\beta^3} \biggl(\frac{x}{\beta}\biggr)^3\ell^4\cdot 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta (1+xb)  
~-~\cancelto{0}{\beta^4}\biggl(\frac{x}{\beta}\biggr)^4 \ell^4\cdot 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta-3] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~\frac{3\cancelto{0}{\beta^4}\ell^4 m^2}{(n+1)}  \biggr[
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2\biggl[ 1-2\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \cancelto{0}{\mathcal{O}(\beta^3)}\biggr] 
+ \biggl[ 1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 6\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \cancelto{0}{\mathcal{O}(\beta^3)} \biggr] \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2 + 1 ] \cdot \biggr[
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+\cancelto{0}{x}b) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^2 2^2\cdot 3 (n+1) 
+ \beta^2 \biggl(\frac{x}{\beta}\biggr)^2 2^2 (n+1)[3 - 2^3(n+1)\cos^2\theta ]\biggl(1+\frac{3\cancelto{0}{x}b}{2}\biggr)^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2+ 1 \biggr\} 
+~\beta^1 \biggl(\frac{x}{\beta}\biggr) \cdot 2^3(n+1)^2\biggl\{4\cos\theta  -4\cos\theta  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~\beta^2 \biggl(\frac{x}{\beta}\biggr)^2 \cdot 2^5(n+1)^2 \cos^2\theta  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2 + 1 ] \cdot \biggr[
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3)  \biggr]
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~\beta^2 2^2\cdot 3 (n+1) \biggl[1 - \biggl(\frac{x}{\beta}\biggr)^2\biggr] \, .</math>
  </td>
</tr>
</table>
</div>

So we see that the coefficients of the lowest-order <math>(\beta^0 ~\mathrm{and} ~ \beta^1)</math> terms are zero, and the coefficient of the <math>~\beta^2</math> term is ''almost'' zero!  My analysis the second time around  gives,


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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \mathrm{Re}\biggl[\mathrm{TERM4} + \mathrm{TERM5}\biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3] 
-~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^2\cdot 3 (n+1)[2^3(n+1)\cos^2\theta -3] b
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta  
+~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^4 (n+1)[2^3(n+1)\cos^2\theta -3] \cos\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+2^3(n+1)^2
\biggl\{ 4\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
-\frac{3\beta^2}{2(n+1)} - 4\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta  
+\frac{2\cdot 3\beta^3}{(n+1)} \biggl(\frac{x}{\beta}\biggr)\cos\theta  \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta] 
+~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2\cdot 3 (n+1) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+2^3(n+1)^2
\biggl\{ 4\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
-\frac{3\beta^2}{2(n+1)} \biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+
\beta^2\biggl(\frac{x}{\beta}\biggr)^2 [2^5(n+1)^2\cos^2\theta ]
-\beta^22^2\cdot 3(n+1)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-\beta^22^2\cdot 3(n+1)\biggl[1-\biggl( \frac{x}{\beta}\biggr)^2 \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

Exactly the same as the first time around.