Editing Appendix/Ramblings/PPToriPt2A (section)

====Imaginary Parts====

=====TERM1=====

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+\frac{1}{x^2} \cdot 
(-1)\beta [2^7\cdot 3 (n+1)^3 ]^{1/2} \biggl\{
(\beta\eta)\cos\theta + \frac{3x^3\sin^2\theta}{2(\beta\eta)}(5\cos^2\theta -2) + \frac{3^2x^6\sin^6\theta\cos\theta}{2^2(\beta\eta)^3}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\beta b_0}{4}  \biggl[ 4b+12\beta\biggl(\frac{x}{\beta}\biggr) b^2\biggr]\biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{-3/2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-\frac{\beta b_0}{2^2x\cos\theta} \biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{1/2}\biggl\{
2^2 \cos\theta + 2\cdot 3 \beta\biggl(\frac{x}{\beta}\biggr) \sin^2\theta (5\cos^2\theta -2)\biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{-1} 
+ 3^2 \beta^2\biggl(\frac{x}{\beta}\biggr)^2 \sin^6\theta\cos\theta \biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{-2}
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

=====TERM2=====

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

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM2}}{\ell^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\beta~\biggl[ \frac{2^5\cdot 3 (n+1)^3}{1+x(3\cos\theta-\cos^3\theta)} \biggr]^{1/2} \biggl\{ 2\cos\theta
- x[2 - 7\cos^2\theta + 3\cos^4\theta ] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~- x^2 \cos\theta [ 9 +4\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>~\frac{\beta b_0}{2\cos\theta}~ \biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{-1/2}\biggl\{ 2\cos\theta
- \beta\biggl(\frac{x}{\beta}\biggr) [2 - 7\cos^2\theta + 3\cos^4\theta ] 
- \beta^2\biggl(\frac{x}{\beta}\biggr)^2 \cos\theta [ 9 +4\cos^2\theta -\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{Im}\biggl[\mathrm{TERM3}\biggr]</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ 
-m^2\beta [  2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-m^2\beta^2 b_0 \biggl(\frac{x}{\beta}\biggr)\biggl[ 1+\beta\biggl(\frac{x}{\beta}\biggr)b \biggr]^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>

=====TERM4=====

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

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \beta\cos\theta [2^5\cdot 3 (n+1)^3]^{1/2} \cdot \frac{x(2+3xb)}{(\beta\eta)}\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>~
-~ \beta \sin\theta [2^7\cdot 3 (n+1)^3 (\beta\eta)^2]^{1/2}\biggl\{ 1 +\frac{3x^3}{2}\cdot\biggl[ \frac{\sin^2\theta \cos\theta}{(\beta\eta)^2} \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>~
-~x \cdot 2\beta\cos\theta [2^7\cdot 3 (n+1)^3]^{1/2} \cdot (1+xb)^{-1/2}\cdot \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^2\cdot 3\beta \sin^4\theta [2^7\cdot 3 (n+1)^3 ]^{1/2} (1+xb)^{1/2} \biggl\{ 1 +\frac{3x}{2}\cdot\biggl[ \frac{\sin^2\theta \cos\theta}{(1+xb)} \biggr]\biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-x\biggl\{~[~109.8335164~] + x[~119.7674436~]~\biggr\}= -34.94384433
</math>
  </td>
</tr>
</table>
</div>

Alternatively we can write,

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

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \beta\cos\theta [2^5\cdot 3 (n+1)^3]^{1/2} \cdot \frac{x(2+3xb)}{(\beta\eta)}\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>~
-~ \beta \sin\theta [2^7\cdot 3 (n+1)^3 (\beta\eta)^2]^{1/2}\biggl\{ 1 +\frac{3x^3}{2}\cdot\biggl[ \frac{\sin^2\theta \cos\theta}{(\beta\eta)^2} \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>~
-2b_0 \beta^2 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
-~ 3b_0\beta^3 \biggl(\frac{x}{\beta}\biggr)^2 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-~ \frac{9b_0}{2} \cdot \beta^4 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta  (1 + xb)^{-1/2}   
</math>
  </td>
</tr>

</table>
</div>


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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\beta^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl\{
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
-~ 3b_0\beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2} 
-~ \frac{9b_0}{2} \cdot \beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta  (1 + xb)^{-1/2}  
\biggr\}  
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ ~-27.45837910~-6.77631589 ~-0.70914934~ \biggr\}\times [~0.58618164~]
=\biggl\{ ~-34.94384433~ \biggr\}\times [~0.58618164~] = -20.48343998
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
-~ 3b_0\beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2} 
-~ \frac{9b_0}{2} \cdot \beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta  (1 + xb)^{-1/2}  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+~
8b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}  \cos\theta
+~ 12b_0\beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2}  \cos\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
-12b_0 \beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} \cos^2\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
+~b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8\biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}  \cos\theta
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\beta^2 b_0\biggl(\frac{x}{\beta}\biggr)^3\biggl\{
-~ \frac{9}{2} \cdot  \sin^6\theta  (1 + xb)^{-1/2}  
+~ 12  \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2}  \cos\theta
-12 \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} \cos^2\theta 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
+~b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos\theta
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

=====TERM5=====

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \cancelto{1}{m^2}
\biggl\{1 - 2\ell^2 + \ell^4 -\frac{3\beta^2\ell^4}{2(n+1)} \biggr\} \cdot 2 \beta x[ ~ 32.12475681~]
+~\sqrt{3}\beta  [\ell^2 -\ell^4] 
\cdot \biggl\{ \biggl[ 2^5 ~- 10\cancelto{1}{m^2}\beta^2\biggr] + 2m^2x^2 \cdot [ ~2.6875~ ] \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \cancelto{1}{m^2}
\biggl\{~-0.38470459~\biggr\} \cdot [ ~16.06237841~]
+~[~0.31080502~] \cdot \biggl\{ 22.3359375\biggr\}= 0.76285080 \, .
</math>
  </td>
</tr>

</table>
</div>


Let's rewrite both of these expressions in terms of a power series in <math>~\beta</math>.
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\mathrm{TERM5}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \beta^2\biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 b_0
\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 + \mathcal{O}(\beta^3) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ + \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]\biggl[1 -\frac{3\beta^2}{2(n+1)} \biggr]\biggr\} \cdot 
\biggl\{ 1 +\beta\biggl(\frac{x}{\beta}\biggr)\frac{b}{2} - \beta^2\biggl(\frac{x}{\beta}\biggr)^2\frac{b^2}{8} + \mathcal{O}(\beta^3)\biggr\}
</math>
  </td>
</tr>

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

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

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

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

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

<tr>
  <td align="right">
<math>~\Rightarrow~~~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 b_0
\biggl\{\beta^0(1-2+1) +4\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta -2 \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta  
-4\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta + 6\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta -\frac{3\beta^2}{2(n+1)}  + \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}{8} + \mathcal{O}(\beta^3)\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~b_0\biggl[ \frac{(1-1)}{\beta\cos\theta} + 2\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta +4\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\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>~
-~ m^2(4n+1)\cdot \biggl[ \frac{2^3\cdot 3}{(n+1)}\biggr]^{1/2} \biggl[ \beta^{1}(1-1) + 2\beta^2\biggl(\frac{x}{\beta}\biggr)\cos\theta 
- 5\beta^3 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta +4\beta^4 \biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta + \mathcal{O}(\beta^5)\biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~m^2 b [2^3(n+1) \cos^2\theta - 3]  \cdot [ 2^3\cdot 3(n+1) ]^{1/2} \biggl[ \beta^2\biggl(\frac{x}{\beta}\biggr)^3 (1-1) + 2\beta^3\biggl(\frac{x}{\beta}\biggr)^4 \cos\theta 
- 5\beta^4\biggl(\frac{x}{\beta}\biggr)^5 \cos^2\theta +4\beta^5\biggl(\frac{x}{\beta}\biggr)^6 \cos^3\theta + \mathcal{O}(\beta^3)\biggr]
</math>
  </td>
</tr>

</table>
</div>

Dropping all terms on the right-hand-side that are <math>~\mathcal{O}(\beta^3)</math> or higher, we have,

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

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 b_0
\biggl\{\beta^0(1-2+1) +(4-4)\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta +4 \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta  
- \beta^2\biggl[ \frac{3}{2(n+1)}\biggr]  + \cancelto{0}{\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}{8} + \cancelto{0}{\mathcal{O}(\beta^3)}\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~b_0\biggl[ \frac{(1-1)}{\beta\cos\theta} + 2\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta +4\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta + \cancelto{0}{\mathcal{O}(\beta^3)}\biggr] 
</math>
  </td>
</tr>

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~m^2 b_0
\biggl\{- \biggl[ \frac{3}{(n+1)}\biggr]\biggl(\frac{x}{\beta}\biggr)   + 8 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta  
\biggr\} \times\biggl\{ \beta^2 +\cancelto{0}{\mathcal{O}(\beta^3)} \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~b_0\biggl[ 2\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta + 4\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta \biggr] 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~2b_0\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5b_0\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta 
+ 4b_0\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+\beta^2 m^2 
\biggl\{- \biggl[ \frac{3b_0}{(n+1)}\biggr]\biggl(\frac{x}{\beta}\biggr)   + 8 b_0\biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta  
-~ (4n+1)\cdot \biggl[ \frac{2^3\cdot 3}{(n+1)}\biggr]^{1/2} \biggl[ 2\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggr] 
+~ [2^3(n+1) \cos^2\theta - 3]  \cdot [ 2^3\cdot 3(n+1) ]^{1/2} \biggl[ 2\biggl( \frac{x}{\beta}\biggr)^3\cos\theta \biggr] \biggr\} \, .
</math>
  </td>
</tr>

</table>
</div>

=====Together=====

Together, then, we have:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}+\mathrm{TERM5}}{b_0\beta^2}\biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2\biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
+ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos\theta
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  \biggr\} 
+ 2\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{x}{\beta}\biggr) (-2 + 2)
-5\beta\biggl(\frac{x}{\beta}\biggr)^2 [3\cos\theta - \cos^3\theta]
+ \frac{3\beta}{\cos\theta} \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ \cos^2\theta
-[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>~
\biggl(\frac{x}{\beta}\biggr) (-2 + 2)
-\frac{5\beta}{\cos\theta}\biggl(\frac{x}{\beta}\biggr)^2 [3\cos^2\theta - \cos^4\theta]
+ \frac{3\beta}{\cos\theta} \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 
-1+3\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>~
\biggl(\frac{x}{\beta}\biggr) (-2 + 2)
+ \frac{\beta}{\cos\theta} \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 
-3+9\cos^2\theta - 3\cos^4\theta -15\cos^2\theta + 5\cos^4\theta \biggr\} 
</math>
  </td>
</tr>

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

</table>
</div>

<!-- Old derivation with algebra errors after the first line
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}+\mathrm{TERM5}}{\beta^2}\biggr]</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2} 
+~b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos\theta
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  \biggr\}
+~\biggl\{2b_0\beta^0\biggl(\frac{x}{\beta}\biggr) 
- 5b_0\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+3xb\biggr) \biggl(1-\frac{xb}{2}\biggr) 
+~2t_5 b_0\beta^0\biggl(\frac{x}{\beta}\biggr)\cos\theta 
+~\frac{b_0}{\cos\theta}~ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos^2\theta
-~ 3 \sin^4\theta  \biggr\}
- 5t_5 b_0\beta\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b_0\biggl\{
-2 \biggl(\frac{x}{\beta}\biggr)  
+~2 \biggl(\frac{x}{\beta}\biggr) (t_5\cos\theta) 
- \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl(6 - 1\biggr) (3\cos\theta -\cos^3\theta)
\biggr\}
+~\frac{b_0}{\cos\theta}~ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos^2\theta
-~ 3 (1 - 2\cos^2\theta + \cos^4\theta) 
- 5 (t_5\cos\theta)\cos^2\theta \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~b_0\biggl\{
-2 \biggl(\frac{x}{\beta}\biggr)  
+~2 \biggl(\frac{x}{\beta}\biggr) (t_5\cos\theta) \biggr\}
+~\frac{b_0}{\cos\theta}~ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos^2\theta
-~ 3 (1 - 2\cos^2\theta + \cos^4\theta) 
- 5 (t_5\cos\theta)\cos^2\theta - \biggl(6 - 1\biggr) (3\cos^2\theta -\cos^4\theta)\biggr\}
</math>
  </td>
</tr>
</table>
</div>
-->

[[File:BetaErrorPlot01.png|center|500px|Beta Error Plot]]

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

<tr>
  <td align="right">
<math>~\mathrm{Im}[\mathrm{TERM4} + \mathrm{TERM5}]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\beta^2 \biggl(\frac{x}{\beta}\biggr) \ell^4 \cos\theta [2^9\cdot 3 (n+1)^3]^{1/2} \cdot (1+xb)^{-1/2}\cdot \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)^2\cdot 3 \ell^4 \sin^4\theta [2^7\cdot 3 (n+1)^3 ]^{1/2} (1+xb)^{1/2} \biggl\{ 1 +\frac{3x}{2}\cdot\biggl[ \frac{\sin^2\theta \cos\theta}{(1+xb)} \biggr]\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)\cdot 2  m^2 [1 - 2\ell^2 + \ell^4 ] 
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (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^4} \biggl(\frac{x}{\beta}\biggr)
\biggl[\frac{3 m^2\ell^4}{(n+1)} \biggr] 
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+xb)^{1/2}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~\cancelto{0}{\beta^3} \biggl[ \frac{2\cdot 3}{(n+1)}\biggr]^{1/2} [\ell^2 -\ell^4] 
\cdot \biggl[ 2m^2(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2 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">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
-~\beta^1 [ 2^7\cdot 3 (n+1)^3]^{1/2} \biggl\{  \biggl[ 1-2\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \cancelto{0}{\mathcal{O}(\beta^2)} \biggr]  
-  \biggl[ 1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \cancelto{0}{\mathcal{O}(\beta^2)} \biggr] \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) \cos\theta [2^9\cdot 3 (n+1)^3]^{1/2} \cdot (1+\cancelto{0}{x}b)^{-1/2}\cdot \biggl(1+\frac{3\cancelto{0}{x}b}{2}\biggr)^2 
</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)\cdot 2  m^2 [1 - 2 + 1 ] 
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+\cancelto{0}{x}b)^{1/2}
</math>
  </td>
</tr>

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