Difference between revisions of "User:Tohline/Appendix/Ramblings/PPToriPt2"

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(→‎Imaginary Parts: Divide into a few subsubsections)
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<!-- __NOTOC__ will force TOC off -->
<!-- __NOTOC__ will force TOC off -->
=Stability Analyses of PP Tori (Part 2)=
=Stability Analyses of PP Tori (Part 2)=
<font color="red"><b>[Comment by J. E. Tohline on 24 May 2016]</b></font> &nbsp; This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura &amp; Hadley collaboration.  I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws.  Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight.  I have written a separate chapter titled, "[[User:Tohline/Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori|Characteristics of Unstable Eigenvectors in Self-Gravitating Tori]]," that contains a much more trustworthy analysis of this very interesting problem.
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{{LSU_HBook_header}}


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===Our Manipulation of These Equations===
===Our Manipulation of These Equations===
 
====Analytic====
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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</div>
</div>


Putting the two together implies,
<span id="KeyExpression">Putting the two together implies,</span>


<div align="center">
<div align="center">
<table border="1" cellpadding="5" align="center"><tr><th align="center">
Definition of Eigenvalue Problem Associated with the Stability of Slim, Papaloizou-Pringle Tori
</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


Line 428: Line 438:


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


The first line of this governing, two-line expression contains the function, <math>~f</math>, as a leading factor, while the leading factor in the second line is the ratio, <math>~n/\beta^2</math>.  Presumably the three terms (hereafter, TERM1, TERM2, &amp; TERM3, respectively) inside the curly brackets on the first line must cancel &#8212; to a sufficiently high order in <math>~x</math> &#8212; and, independently, the two terms (hereafter, TERM4 &amp; Term5, respectively) inside the curly brackets on the second line must cancel.  Furthermore, these cancellations must occur separately for the real parts and the imaginary parts of each bracketed expression.
The first line of this governing, two-line expression contains the function, <math>~f</math>, as a leading factor, while the leading factor in the second line is the ratio, <math>~n/\beta^2</math>.  Presumably the three terms (hereafter, TERM1, TERM2, &amp; TERM3, respectively) inside the curly brackets on the first line must cancel &#8212; to a sufficiently high order in <math>~x</math> &#8212; and, independently, the two terms (hereafter, TERM4 &amp; Term5, respectively) inside the curly brackets on the second line must cancel.  Furthermore, these cancellations must occur separately for the real parts and the imaginary parts of each bracketed expression.


====Example Evaluation====
Evaluating various terms using the parameter set, &nbsp;&nbsp;
Evaluating various terms using the parameter set, &nbsp;&nbsp;
<math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>
<math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>
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<tr>
<tr>
   <td align="right">
   <td align="right">
TERM4
TERM1
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] </math>
-x \ell^4\biggl[ (2+3xb)\cdot \frac{\partial\Lambda}{\partial x} -  3\sin^3\theta \cdot \frac{\partial\Lambda}{\partial \theta}   \biggr]
</math>
   </td>
   </td>
</tr>
</tr>
Line 461: Line 471:
   <td align="left">
   <td align="left">
<math>~
<math>~
-x \ell^4\biggl[ (2+3xb)\cdot [~1.515625000~\pm~i~36.23373732 ~\beta]
\biggl(\frac{7}{2^3} \biggr)^2\biggl\{ \frac{65}{2^3} + \frac{1}{2^4}\cdot [~4.269531250~] \biggr\}
3\sin^3\theta \cdot [~-2.388335684~\pm~i~(-1)15.36617018 ~\beta]   \biggr]
~\pm~i~\biggl(\frac{7}{2^3} \biggr)^2\biggl\{ [~30.76957507~] + \frac{1}{2^4}\cdot  (-1)[~5.773638858~] \biggr\}\beta
</math>
</math>
   </td>
   </td>
Line 476: Line 486:
   <td align="left">
   <td align="left">
<math>~
<math>~
-x\ell^4 [~9.248046874~\pm~i~139.7753772~\beta]
\frac{7^2}{2^6} [ ~8.39184570
~\pm~i~30.40872264~\beta] \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
TERM5 (Case B)
TERM2
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x}
\biggl[ \ell^4 [1-0.75\beta^2~\pm~i~(-1)\sqrt{3}\beta]
+ \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] </math>
+2\ell^2[ -1~\pm~i~\sqrt{0.75}\beta ] + 1 \biggr]
\cdot \biggl[~2^5 + 2\cancelto{1}{m^2}[~- 5\beta + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr]
</math>
   </td>
   </td>
</tr>
</tr>
Line 511: Line 514:
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]
\frac{7}{2^5} [ ~-0.931640625
\cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]~\pm~i~[(2)8.031189202 ~\beta]~\biggr]
~\pm~i~13.86780926~\beta] \, .
</math>
</math>
   </td>
   </td>
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<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
TERM3
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~- [ 2^2(n+1)^2 + m^2\Lambda ] </math>
\pm~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[~i~ [2^5 - 10\beta + (2)0.167968750]~-~[(2)8.031189202 ~\beta]~\biggr]
</math>
   </td>
   </td>
</tr>
</tr>
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   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]
-\biggl\{~2^4 + m^2[~- 5\beta^2 + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr\}\, .
\cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]\biggr]
\pm~(-1)\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[[(2)8.031189202 ~\beta]~\biggr]
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\pm~i~\biggl\{\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]
\cdot \biggl[[(2)8.031189202 ~\beta]~\biggr]
+~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[~ [2^5 - 10\beta + (2)0.167968750]~\biggr] \biggr\} \, .
</math>
</math>
   </td>
   </td>
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</div>
</div>


Evaluating this TERM5 expression for the case of <math>~\beta = 1</math>, we have,
The sum of these three terms gives,
 


<div align="center">
<div align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
TERM5 (Case B)
TERM1 + TERM2 + TERM3
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~  
\biggl[ 0.25\ell^4 - 2\ell^2 + 1 \biggr]  
\frac{7^2}{2^6} [ ~8.39184570
\cdot \biggl[~[2^5 - 10 + (2)0.167968750]\biggr]
~\pm~i~30.40872264~\beta]  
\pm~(-1)\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr]
+\frac{7}{2^5} [ ~-0.931640625
\cdot \biggl[[(2)8.031189202 ]~\biggr]
~\pm~i~13.86780926~\beta]  
</math>
</math>
   </td>
   </td>
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<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~  
\pm~i~\biggl\{\biggl[ \ell^4 [1-0.75] - 2\ell^2 + 1 \biggr]
-\biggl\{~2^4 + m^2[~- 5\beta^2 + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr\}
\cdot \biggl[[(2)8.031189202]~\biggr]
+~\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[~ [2^5 - 10 + (2)0.167968750]~\biggr] \biggr\}  
</math>
</math>
   </td>
   </td>
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<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;  
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~  
[ -0.38470459 ]
6.42500686 - 0.20379639
\cdot [22.3359375]
-~2^4 + 5m^2\beta^2 - m^2 0.167968750
\pm~(-1)[ ~0.31080502 ]
\cdot [~16.0623784 ~]
</math>
</math>
   </td>
   </td>
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<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;  
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 633: Line 607:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~ \pm~i~\biggl[23.28167827 + 3.03358328
\pm~i~\biggl\{[ -0.38470459 ]
- 8.031189202 ~m^2~\biggr]\beta
\cdot [~16.0623784 ~]
+~[ ~0.31080502 ] 
\cdot [22.3359375]\biggr\}
</math>
</math>
   </td>
   </td>
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<tr>
<tr>
  <td align="right">
&nbsp;
  </td>
   <td align="center">
   <td align="center">
&nbsp;
  </td>
  <td align="right">
<math>~=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~[~-13.58500545~]
<math>~  
\pm~i~[~0.76285080~] \, .
-9.77878953+ 5m^2\beta^2 - m^2 0.167968750
~ \pm~i~\biggl[26.31526155- 8.031189202 ~m^2\biggr]\beta
</math>
</math>
   </td>
   </td>
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</div>
</div>


===Testing for Expected Cancellations===
Moving on to the last pair of terms &hellip;
Note first that, adopting the shorthand notation,


<div align="center">
<div align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\ell</math>
TERM4
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~(1-x\cos\theta)</math>
<math>~
-x \ell^4\biggl[ (2+3xb)\cdot \frac{\partial\Lambda}{\partial x} - 3\sin^3\theta \cdot \frac{\partial\Lambda}{\partial \theta}  \biggr]
</math>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow ~~~~\ell^2</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~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) \, ;</math>
<math>~
-x \ell^4\biggl[ (2+3xb)\cdot [~1.515625000~\pm~i~36.23373732 ~\beta]
-  3\sin^3\theta \cdot [~-2.388335684~\pm~i~(-1)15.36617018 ~\beta\biggr]
</math>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\ell^3</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~1-3\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 3\beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \, ;</math>
<math>~
  </td>
-x\ell^4 [~9.248046874~\pm~i~139.7753772~\beta]
</tr>
</math>
 
<tr>
  <td align="right">
<math>~\ell^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~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) \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
====Real Parts====


<div align="center">
<div align="center">
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
TERM5 (Case B)
   </td>
   </td>
   <td align="center">
   <td align="center">
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   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl\{ (n+1)[2^3(n+1)\cos^2\theta -3]x(2+3xb)\biggr\} \cdot \biggl[ -x(2+3xb) \biggr]
\biggl[ \ell^4 [1-0.75\beta^2~\pm~i~(-1)\sqrt{3}\beta]
+2\ell^2[  -1~\pm~i~\sqrt{0.75}\beta ] + 1 \biggr]
\cdot \biggl[~2^5 + 2\cancelto{1}{m^2}[~- 5\beta + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr]
</math>
</math>
   </td>
   </td>
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   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
+~ (n+1)\sin\theta \biggl\{
\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]  
-2^4 (n+1) (\beta\eta)^2 \cos\theta + 3x^3 \sin^2\theta \biggl[3 - 2^3(n+1)\cos^2\theta \biggr]
\cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]~\pm~i~[(2)8.031189202 ~\beta]~\biggr]
\biggr\} \cdot \biggl[ 3x\sin^3\theta \biggr]  
</math>
</math>
   </td>
   </td>
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   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~(n+1)[2^3(n+1)\cos^2\theta -3]x^2(2+3xb)^2
\pm~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]  
\cdot \biggl[~i~ [2^5 - 10\beta + (2)0.167968750]~-~[(2)8.031189202 ~\beta]~\biggr]
</math>
</math>
   </td>
   </td>
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   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~ 3x^3(n+1)\sin^4\theta \biggl\{
\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]
2^4 (n+1) (1+xb) \cos\theta + 3x \sin^2\theta [2^3(n+1)\cos^2\theta -3]
\cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]\biggr]
\biggr\} 
\pm~(-1)\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[[(2)8.031189202 ~\beta]~\biggr]
</math>
</math>
   </td>
   </td>
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   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~x^2 \cdot 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]\biggl(1+\frac{3xb}{2}\biggr)^2
\pm~i~\biggl\{\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr]
\cdot \biggl[[(2)8.031189202 ~\beta]~\biggr]
+~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[~ [2^5 - 10\beta + (2)0.167968750]~\biggr] \biggr\} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Evaluating this TERM5 expression for the case of <math>~\beta = 1</math>, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
TERM5 (Case B)
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~ x^3 \cdot 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta (1+xb)
\biggl[ 0.25\ell^4 - 2\ell^2 + 1 \biggr]
~-~x^4\cdot 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta -3] \, .</math>
\cdot \biggl[~[2^5 - 10 + (2)0.167968750]\biggr]
\pm~(-1)\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr]
\cdot \biggl[[(2)8.031189202 ]~\biggr]
</math>
   </td>
   </td>
</tr>
</tr>
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   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-x\biggl\{~x[~18.37695315~] + x^2[~72.5625~] + x^3[~7.59375~]~~\biggr\} = -x[~9.24804688~]\, .
\pm~i~\biggl\{\biggl[ \ell^4 [1-0.75] - 2\ell^2 + 1 \biggr]
\cdot \biggl[[(2)8.031189202]~\biggr]
+~\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr]  
\cdot \biggl[~ [2^5 - 10 + (2)0.167968750]~\biggr] \biggr\}
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Re}\biggl[\mathrm{TERM5}\biggr]</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 836: Line 814:
   <td align="left">
   <td align="left">
<math>~
<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 ]  
[ -0.38470459 ]  
-\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 ]  
\cdot [22.3359375]
\pm~(-1)[ ~0.31080502 ]  
\cdot [~16.0623784 ~]
</math>
</math>
   </td>
   </td>
Line 844: Line 824:
<tr>
<tr>
   <td align="right">
   <td align="right">
<b><font color="red" size="+1">Case B:</font></b>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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
\pm~i~\biggl\{[ -0.38470459 ]  
+ 2m^2\biggl[ ~- (4n+1)\beta^2 + (n+1)^2(2^3 \cos^2\theta - 3) x^2(1+xb)\biggr] \biggr\}
\cdot [~16.0623784 ~]
+~[ ~0.31080502 ] 
\cdot [22.3359375]\biggr\}  
</math>
</math>
   </td>
   </td>
Line 858: Line 840:


<tr>
<tr>
   <td align="right">
   <td align="center">
&nbsp;
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="right">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~[~-13.58500545~]
-~\biggl\{ \ell^4(-1)\biggl[\frac{2\cdot 3\beta^2}{(n+1)}\biggr]^{1/2}
\pm~i~[~0.76285080~] \, .
+ 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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
===Testing for Expected Cancellations===
Note first that, adopting the shorthand notation,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\ell</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~(1-x\cos\theta)</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>
   </td>
</tr>
</tr>
Line 889: Line 875:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\Rightarrow ~~~~\ell^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~1-2\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \, ;</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>
   </td>
</tr>
</tr>
</table>
</div>
When added together, we obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Re}[\mathrm{TERM4} + \mathrm{TERM5}]</math>
<math>~\ell^3</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 915: Line 893:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~1-3\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 3\beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \, ;</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>
   </td>
</tr>
</tr>
Line 923: Line 899:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\ell^4</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 6\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta  
-~ \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) 
- 4\beta^3\biggl(\frac{x}{\beta}\biggr)^3\cos^3\theta + \beta^4\biggl(\frac{x}{\beta}\biggr)^4\cos^4\theta \, .</math>
~-~\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>
   </td>
</tr>
</tr>
</table>
</div>
====Real Parts====
=====TERM1=====
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{\ell^2}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
+~\biggl\{1 - 2\ell^2 + \ell^4 \biggr\} \cdot \biggl\{ 2^3(n+1)^2 + 2m^2\beta^2\biggr[
2(n+1)[2^3(n+1)\cos^2\theta -3](1+3xb)  
- (4n+1) + \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]\biggr\}
+2^4(n+1)^2(\sin^2\theta - \cos^2\theta)
</math>
</math>
   </td>
   </td>
Line 959: Line 943:
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\frac{3\beta^2\ell^4}{2(n+1)} \biggl\{ 2^3(n+1)^2 + 2m^2\beta^2\biggr[
+ \beta\biggl(\frac{x}{\beta}\biggr) \biggl[
- (4n+1) + \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]\biggr\}
-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>
</math>
   </td>
   </td>
Line 970: Line 955:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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}
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>
</math>
   </td>
   </td>
Line 984: Line 971:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
\beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2\ell^2 + \ell^4 \biggr\}
+ \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>
</math>
   </td>
   </td>
Line 998: Line 987:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2\ell^2 + \ell^4 ] \cdot \biggr[
-6(n+1)
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]
+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>
</math>
   </td>
   </td>
Line 1,017: Line 1,007:
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2\ell^4 2^2\cdot 3 (n+1)  
-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)
+ \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
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,028: Line 1,018:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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}
-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>
</math>
   </td>
   </td>
Line 1,045: Line 1,037:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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)
-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  
~-~\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>
  + 2^3\cdot 23 (n+1)^2 \cos^4\theta
\biggr\}
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,056: Line 1,050:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
+~\frac{3\cancelto{0}{\beta^4}\ell^4 m^2}{(n+1)\biggr[
-6(n+1)
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+xb) \biggr]
+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>
</math>
   </td>
   </td>
Line 1,071: Line 1,066:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~ + \cos^2\theta\biggl[2^3(n+1)^2(~18 -23 -6~) + 3^2(n+1)(~2-16n-19~)
\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]  
\biggr] + 2^3(n+1)^2\cos^4\theta\biggl[ - 2\cdot 3 + 2 + 23  \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\}  
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,086: Line 1,081:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2 + 1 ] \cdot \biggr[
-6(n+1)
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3) (1+\cancelto{0}{x}b) \biggr]
+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>
</math>
   </td>
   </td>
Line 1,104: Line 1,100:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~ - \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
-~\beta^2 2^2\cdot 3 (n+1)
\biggr\}
+ \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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{(n+1)}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
\beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2+ 1 \biggr\}
\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]
+~\beta^1 \biggl(\frac{x}{\beta}\biggr) \cdot 2^3(n+1)^2\biggl\{4\cos\theta  -4\cos\theta  \biggr\}
</math>
</math>
   </td>
   </td>
Line 1,134: Line 1,133:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
+~\beta^2 \biggl(\frac{x}{\beta}\biggr)^2 \cdot 2^5(n+1)^2 \cos^2\theta   
+\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>
</math>
   </td>
   </td>
Line 1,145: Line 1,146:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2 \cdot 2m^2 [ 1 - 2 + 1 ] \cdot \biggr[
\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr]
(4n+1) - \biggl(\frac{x}{\beta}\biggr)^2(n+1)^2(2^3 \cos^2\theta - 3)  \biggr]
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl[ 12 - 2^5(n+1) + 2^5(n+1)\cos^2\theta\biggr]  
</math>
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,163: Line 1,164:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2 2^2\cdot 3 (n+1) \biggl[1 - \biggl(\frac{x}{\beta}\biggr)^2\biggr]
+\beta\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggl\{ 3^2(16n +13)
- \beta^2 \biggl(\frac{x}{\beta}\biggr)^2 [2^5(n+1)^2\cos^2\theta ]
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,175: Line 1,177:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~-~\beta^2 2^2\cdot 3 (n+1) \biggl[1 - \biggl(\frac{x}{\beta}\biggr)^2\biggr] \, .</math>
<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>
   </td>
</tr>
</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!
====Imaginary Parts====
=====TERM4=====
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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]
- 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>
</math>
   </td>
   </td>
Line 1,214: Line 1,210:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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]
+\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>
</math>
   </td>
   </td>
Line 1,229: Line 1,227:
   <td align="left">
   <td align="left">
<math>~
<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
\biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr]
</math>
</math>
   </td>
   </td>
Line 1,242: Line 1,240:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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\}  
+\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>
</math>
   </td>
   </td>
Line 1,253: Line 1,253:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-x\biggl\{~[~109.8335164~] + x[~119.7674436~]~\biggr\}= -34.94384433
- 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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


Alternatively we can write,
<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">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 1,271: Line 1,288:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{\ell}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,278: Line 1,295:
   <td align="left">
   <td align="left">
<math>~
<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]
-6(n+1) + 2^4(n+1)^2\cos^2\theta
</math>
</math>
   </td>
   </td>
Line 1,292: Line 1,309:
   <td align="left">
   <td align="left">
<math>~
<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]
- \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>
</math>
   </td>
   </td>
Line 1,302: Line 1,319:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \beta^2 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
+\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
-~ 3b_0\beta^3 \biggl(\frac{x}{\beta}\biggr)^2 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2}  
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,317: Line 1,334:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~(n+1)\biggl[ -6 + 2^4(n+1)\cos^2\theta \biggr]
-~ \frac{9b_0}{2} \cdot \beta^4 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta (1 + xb)^{-1/2} 
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow ~~~ \mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\beta^2}\biggr]</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl\{
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
- \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\}
-~ 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>
</math>
   </td>
   </td>
Line 1,359: Line 1,365:
   <td align="left">
   <td align="left">
<math>~
<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
+\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
+ \biggl(\frac{x}{\beta}\biggr)^4\cos^4\theta \biggr\}  
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,367: Line 1,373:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~</math>
<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{(n+1)}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,374: Line 1,380:
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl\{ ~-27.45837910~-6.77631589 ~-0.70914934~ \biggr\}\times [~0.58618164~]
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
=\biggl\{ ~-34.94384433~ \biggr\}\times [~0.58618164~] = -20.48343998
</math>
</math>
   </td>
   </td>
Line 1,385: Line 1,390:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
- \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\}
-~ 3b_0\beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  (1 + xb)^{1/2}
\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
-~ \frac{9b_0}{2} \cdot \beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta (1 + xb)^{-1/2} 
</math>
</math>
   </td>
   </td>
Line 1,404: Line 1,408:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+~
<math>~
8b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2\cos\theta
+\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
+~ 12b_0\beta^2 \biggl(\frac{x}{\beta}\biggr)^3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr] (1 + xb)^{1/2}  \cos\theta
\biggr\}\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr]
</math>
</math>
   </td>
   </td>
Line 1,414: Line 1,418:
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;
  </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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  
\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>
</math>
   </td>
   </td>
Line 1,430: Line 1,434:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
- \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\}
+~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>
</math>
   </td>
   </td>
Line 1,449: Line 1,451:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+\beta^2 b_0\biggl(\frac{x}{\beta}\biggr)^3\biggl\{
<math>~
-~ \frac{9}{2} \cdot  \sin^6\theta (1 + xb)^{-1/2} 
+ \beta^2\biggl(\frac{x}{\beta}\biggr)^\biggl\{ [ 31 + 16n ]\cos^2\theta - [65 + 56n] \cos^4\theta +2^3\cdot 3(n+1)\cos^6\theta \biggr\}
+~ 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>
</math>
   </td>
   </td>
Line 1,463: Line 1,462:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
+\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
+~b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos\theta
\biggr\}
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  \biggr\} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
=====TERM5=====
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}\biggl[\mathrm{TERM5}\biggr]</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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 ]
-\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
+\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 ]
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,499: Line 1,489:
<tr>
<tr>
   <td align="right">
   <td align="right">
<b><font color="red" size="+1">Case B:</font></b>
&nbsp;
  </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ x\cdot 2 \beta m^2
<math>~
\biggl\{1 - 2\ell^2 + \ell^4 -\frac{3\beta^2\ell^4}{2(n+1)} \biggr\}
\biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+xb)^{1/2}
</math>
</math>
   </td>
   </td>
Line 1,521: Line 1,509:
   <td align="left">
   <td align="left">
<math>~
<math>~
+~\beta \biggl[ \frac{2\cdot 3}{(n+1)}\biggr]^{1/2} [\ell^2 -\ell^4]
- \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\}
\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>
</math>
   </td>
   </td>
Line 1,532: Line 1,519:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \cancelto{1}{m^2}
<math>~
\biggl\{1 - 2\ell^2 + \ell^4 -\frac{3\beta^2\ell^4}{2(n+1)} \biggr\} \cdot 2 \beta x[ ~ 32.12475681~]
+ \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\}
+~\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>
</math>
   </td>
   </td>
Line 1,548: Line 1,533:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \cancelto{1}{m^2}
<math>~
\biggl\{~-0.38470459~\biggr\} \cdot [ ~16.06237841~]
-\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
+~[~0.31080502~] \cdot \biggl\{ 22.3359375\biggr\}= 0.76285080 \, .
\biggr\}
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>




Let's rewrite both of these expressions in terms of a power series in <math>~\beta</math>.
=====Sum of TERM1 and TERM2=====
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 1,568: Line 1,552:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}\biggl[\mathrm{TERM5}\biggr]</math>
<math>~
\mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2}}{(n+1)} \biggr]
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,574: Line 1,560:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \beta^2\biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 b_0
<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 + \mathcal{O}(\beta^3) \biggr]  
\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>
</math>
   </td>
   </td>
Line 1,588: Line 1,574:
   </td>
   </td>
   <td align="left">
   <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
<math>~
\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\}
+ \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>
</math>
   </td>
   </td>
Line 1,603: Line 1,589:
   <td align="left">
   <td align="left">
<math>~
<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
+ \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\}
- 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>
</math>
   </td>
   </td>
Line 1,618: Line 1,602:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{(136n +112)  
- 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]  
- \cos^2\theta\biggl[224n + 233  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\cdot \biggl\{ ~- 2m^2(4n+1)\beta^2 \biggr\}
\biggr\}  
</math>
</math>
   </td>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="right">
   <td align="right">
Line 1,635: Line 1,618:
   <td align="left">
   <td align="left">
<math>~
<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  
-\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
- 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]  
\biggr\}
\cdot \biggl\{ x^2 \cdot 2m^2(n+1)[2^3(n+1) \cos^2\theta - 3]  \biggr\}
</math>
</math>
   </td>
   </td>
Line 1,650: Line 1,632:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13)
- 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]  
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\cdot \biggl\{ x^3 b \cdot 2m^2(n+1)[2^3(n+1) \cos^2\theta - 3] \biggr\}
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,659: Line 1,641:
</table>
</table>
</div>
</div>
=====TERM3=====


<div align="center">
<div align="center">
Line 1,665: Line 1,650:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>
<math>~\mathrm{Re}\biggl[\mathrm{TERM3}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 1,671: Line 1,656:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \biggl(\frac{x}{\beta}\biggr)\cdot 2 m^2 b_0
<math>~-  2^2(n+1)^2 +
\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 
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]  
-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>
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,686: Line 1,670:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \times
<math>~
\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\}
- 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>
</math>
   </td>
   </td>
Line 1,701: Line 1,699:
   <td align="left">
   <td align="left">
<math>~
<math>~
+~b_0\biggl[ \frac{(1-1)}{\beta\cos\theta} + 2\beta^0\biggl(\frac{x}{\beta}\biggr)  
- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
=====Sum of TERM1 + TERM2 + TERM3=====
Therefore,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~
\mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}}{(n+1)} \biggr]
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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
\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)  
- 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>
</math>
   </td>
   </td>
Line 1,731: Line 1,736:
   <td align="left">
   <td align="left">
<math>~
<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
+ \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\}
- 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>
</math>
   </td>
   </td>
Line 1,746: Line 1,750:
   <td align="left">
   <td align="left">
<math>~
<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  
+ \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\}
- 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>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</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">
   <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \biggl(\frac{x}{\beta}\biggr)\cdot 2 m^2 b_0
<math>~   
\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
- 2\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta   \biggl\{(136n +112)  
- \beta^2\biggl[ \frac{3}{2(n+1)}\biggr]  + \cancelto{0}{\mathcal{O}(\beta^3)} \biggr\}  
- \cos^2\theta\biggl[224n + 233  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}  
</math>
</math>
   </td>
   </td>
Line 1,783: Line 1,779:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \times
<math>
\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\}
+ 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>
</math>
   </td>
   </td>
Line 1,798: Line 1,794:
   <td align="left">
   <td align="left">
<math>~
<math>~
+~b_0\biggl[ \frac{(1-1)}{\beta\cos\theta} + 2\beta^0\biggl(\frac{x}{\beta}\biggr)  
-\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
- 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]
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,812: Line 1,808:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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  
+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13)  
+ \cancelto{0}{\mathcal{O}(\beta^3)}\biggr]
- \cos^2\theta\biggl[232n + 241  \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
\biggr\}
</math>
</math>
   </td>
   </td>
Line 1,828: Line 1,825:
   <td align="left">
   <td align="left">
<math>~
<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  
- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr]
+ \cancelto{0}{\mathcal{O}(\beta^3)}\biggr]  
</math>
</math>
   </td>
   </td>
Line 1,839: Line 1,835:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~12n
+~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]
+ \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>
</math>
   </td>
   </td>
</tr>
</tr>


<tr>
</table>
   <td align="right">
</div>
&nbsp;
 
=====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>
   <td align="center">
   <td align="center">
<math>~\approx</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~m^2 b_0
<math>~
\biggl\{- \biggl[ \frac{3}{(n+1)}\biggr]\biggl(\frac{x}{\beta}\biggr)  + 8 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta
\biggl\{ (n+1)[2^3(n+1)\cos^2\theta -3]x(2+3xb)\biggr\} \cdot \biggl[ -x(2+3xb) \biggr]
\biggr\} \times\biggl\{ \beta^2 +\cancelto{0}{\mathcal{O}(\beta^3)} \biggr\}
</math>
</math>
   </td>
   </td>
Line 1,872: Line 1,874:
   <td align="left">
   <td align="left">
<math>~
<math>~
+~b_0\biggl[ 2\beta^0\biggl(\frac{x}{\beta}\biggr)
+~ (n+1)\sin\theta \biggl\{
- 5\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta + 4\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta \biggr]  
-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>
</math>
   </td>
   </td>
Line 1,883: Line 1,886:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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]
-~(n+1)[2^3(n+1)\cos^2\theta -3]x^2(2+3xb)^2
</math>
</math>
   </td>
   </td>
Line 1,901: Line 1,904:
   <td align="left">
   <td align="left">
<math>~
<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]
-~ 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>
</math>
   </td>
   </td>
Line 1,911: Line 1,916:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~2b_0\beta^0\biggl(\frac{x}{\beta}\biggr)  
<math>~
- 5b_0\beta\biggl(\frac{x}{\beta}\biggr)^2\cos\theta  
-~x^2 \cdot 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]\biggl(1+\frac{3xb}{2}\biggr)^2
+ 4b_0\beta^2 \biggl(\frac{x}{\beta}\biggr)^3\cos^2\theta
</math>
</math>
   </td>
   </td>
Line 1,929: Line 1,933:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+\beta^2 m^2
<math>~
\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  
-~ x^3 \cdot 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta (1+xb)   
-~ (4n+1)\cdot \biggl[ \frac{2^3\cdot 3}{(n+1)}\biggr]^{1/2} \biggl[ 2\biggl(\frac{x}{\beta}\biggr)\cos\theta \biggr]
~-~x^4\cdot 3^2(n+1)\sin^6\theta [2^3(n+1)\cos^2\theta -3] \, .</math>
+~ [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\} \, .
  </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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>


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


Together, then, we have:
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 1,948: Line 1,962:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}+\mathrm{TERM5}}{\beta^2}\biggr]</math>
<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+\frac{3xb}{2} \biggr)^2 (1+xb)^{-1/2}
-~\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]
+~b_0 \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos\theta
-~ 3 \biggl[\frac{\sin^4\theta}{\cos\theta}\biggr]  \biggr\}
+~t_5 \times \biggl\{2b_0\beta^0\biggl(\frac{x}{\beta}\biggr)\cos\theta
- 5b_0\beta\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr\}
</math>
</math>
   </td>
   </td>
Line 1,969: Line 1,979:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-2b_0 \biggl(\frac{x}{\beta}\biggr) \biggl(1+3xb\biggr) \biggl(1-\frac{xb}{2}\biggr)
-~ \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] 
+~2t_5 b_0\beta^0\biggl(\frac{x}{\beta}\biggr)\cos\theta
~-~\beta^4 \biggl( \frac{x}{\beta}\biggr)^4 3^2(n+1)\sin^6\theta  [2^3(n+1)\cos^2\theta -3] </math>
+~\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>
   </td>
</tr>
</tr>
Line 1,987: Line 1,993:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\approx</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~b_0\biggl\{
<math>~
-2 \biggl(\frac{x}{\beta}\biggr)
-~\beta^2 \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]
+~2 \biggl(\frac{x}{\beta}\biggr) (t_5\cos\theta)
-~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^2\cdot 3 (n+1)[2^3(n+1)\cos^2\theta -3] b
- \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl(6 - 1\biggr) (3\cos\theta -\cos^3\theta)
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta
\biggr\}
</math>
+~\frac{b_0}{\cos\theta}~ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos^2\theta
  </td>
-~ 3 (1 - 2\cos^2\theta + \cos^4\theta)
</tr>
- 5 (t_5\cos\theta)\cos^2\theta \biggr\}
 
<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>
</math>
   </td>
   </td>
Line 2,007: Line 2,024:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~b_0\biggl\{
<math>~
-2 \biggl(\frac{x}{\beta}\biggr)
-~ \beta^3\biggl( \frac{x}{\beta}\biggr)^3 2^4\cdot 3(n+1)^2 \cos\theta\sin^4\theta 
+~2 \biggl(\frac{x}{\beta}\biggr) (t_5\cos\theta) \biggr\}
+~\beta^3 \biggl( \frac{x}{\beta}\biggr)^3 2^4 (n+1)[2^3(n+1)\cos^2\theta -3] \cos\theta
+~\frac{b_0}{\cos\theta}~ \beta \biggl(\frac{x}{\beta}\biggr)^2 \biggl\{ 8 \cos^2\theta
</math> \, .
-~ 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>
   </td>
</tr>
</tr>
Line 2,022: Line 2,036:
</div>
</div>


 
=====TERM5=====
{{LSU_WorkInProgress}}
Now, let's examine the TERM5 expressions.
When added together, we obtain,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 2,030: Line 2,043:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathrm{Im}[\mathrm{TERM4} + \mathrm{TERM5}]</math>
<math>~\mathrm{Re}\biggl[\mathrm{TERM5}\biggr]</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 2,037: Line 2,050:
   <td align="left">
   <td align="left">
<math>~
<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  
\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>
</math>
   </td>
   </td>
Line 2,044: Line 2,058:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<b><font color="red" size="+1">Case B:</font></b>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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\}  
\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>
</math>
   </td>
   </td>
Line 2,064: Line 2,079:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+\beta^2 \biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 [1 - 2\ell^2 + \ell^4 ]
<math>~
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+xb)^{1/2}
-~\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>
</math>
   </td>
   </td>
Line 2,075: Line 2,091:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~-\cancelto{0}{\beta^4} \biggl(\frac{x}{\beta}\biggr)
<math>~
\biggl[\frac{3 m^2\ell^4}{(n+1)} \biggr]
\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
\cdot [ 2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+xb)^{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>
</math>
   </td>
   </td>
Line 2,094: Line 2,110:
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta [ 2^7\cdot 3 (n+1)^3]^{1/2} [\ell^2 -\ell^4]
-~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>
</math>
   </td>
   </td>
Line 2,104: Line 2,120:
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
+~\cancelto{0}{\beta^3} \biggl[ \frac{2\cdot 3}{(n+1)}\biggr]^{1/2} [\ell^2 -\ell^4]  
\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]  
\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]  
+ \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>
</math>
   </td>
   </td>
Line 2,119: Line 2,136:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<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] 
-\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  
-  \biggl[ 1-4\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \cancelto{0}{\mathcal{O}(\beta^2)} \biggr] \biggr\}
+ \mathcal{O}(\beta^3) \biggr] \biggr\}  
</math>
</math>
   </td>
   </td>
Line 2,137: Line 2,154:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\times
-~\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  
\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>
</math>
   </td>
   </td>
Line 2,151: Line 2,170:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+\beta^2 \biggl(\frac{x}{\beta}\biggr)\cdot m^2 [1 - 2 + 1 ]
<math>~
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta \cdot (1+\cancelto{0}{x}b)^{1/2}
-~\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>
</math>
   </td>
   </td>
Line 2,162: Line 2,181:
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
&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>
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^1 [ 2^7\cdot 3 (n+1)^3]^{1/2} [1 - 1]
\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>
</math>
   </td>
   </td>
Line 2,180: Line 2,216:
   <td align="left">
   <td align="left">
<math>~
<math>~
-~\beta^2 \biggl(\frac{x}{\beta}\biggr)\cos\theta  [ 2^9\cdot 3 (n+1)^3]^{1/2
-\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  
-~\beta^2 \biggl(\frac{x}{\beta}\biggr) \cos\theta [2^9\cdot 3 (n+1)^3]^{1/2}  
+ \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>
</math>
   </td>
   </td>
Line 2,194: Line 2,245:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~+\beta^2 \biggl(\frac{x}{\beta}\biggr)\cdot 2  m^2 [1 - 2 + 1 ]  
<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 TERM$ 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.
 
====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]]
 
{{LSU_WorkInProgress}}
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  
\cdot [  2^7\cdot 3(n+1)^3 ]^{1/2} \cos\theta  
</math>
</math>
Line 2,201: Line 4,053:
</table>
</table>
</div>
</div>
===Summary===
[[#KeyExpression|As stated above]], the eigenvalue problem that must be solved in order to identify the eigenfunction, <math>~\Lambda(x,\theta)</math>, and eigenfrequency, <math>~(\nu/m)</math>, of unstable (as well as stable) nonaxisymmetric modes in slim <math>~(\beta \ll 1)</math>, polytropic <math>~(n)</math> PP tori with uniform specific angular momentum is defined by the following two-dimensional <math>~(x,\theta)</math>, 2<sup>nd</sup>-order PDE:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~f (1-x\cos\theta)^2 \biggl\{
~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}
\biggr\}
+ ~\frac{n}{\beta^2} \biggl\{ \mathrm{TERM4}
~+~ \mathrm{TERM5}\biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
where, <math>~f(x,\theta)</math> is the enthalpy distribution in the unperturbed, axisymmetric torus, and
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathrm{TERM1}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathrm{TERM2}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x}
+ \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathrm{TERM3}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~- [ 2^2(n+1)^2 + m^2\Lambda ] \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathrm{TERM4}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x\cos\theta)^4\biggl[ \frac{\partial \Lambda}{\partial x} \cdot \frac{\partial (\beta^2 f)}{\partial x}
~+~ \frac{\partial \Lambda}{\partial \theta} \cdot \frac{\partial (\beta^2 f/x^2)}{\partial \theta} \biggr] \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathrm{TERM5}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr]
[ 2^3(n+1)^2 + 2m^2\Lambda ] \, .</math>
  </td>
</tr>
</table>
</div>
We also should appreciate that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f\ell^2 \equiv f(1-x\cos\theta)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1-\eta^2)(1-2x\cos\theta + x^2\cos^2\theta)</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 - \beta\biggl(\frac{x}{\beta}\biggr)^3 b\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[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr]
-~\beta\biggl(\frac{x}{\beta}\biggr)^3 b
\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[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
\biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr] + \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>
If an exact solution, <math>~(\Lambda,\nu/m)</math>, to this eigenvalue problem were plugged into this governing PDE, we would expect that ''both'' of the following summations would be exactly zero at all meridional-plane <math>~(x,\theta)</math> locations throughout the torus:
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\mathrm{TERM4} + \mathrm{TERM5}  \, .</math>
  </td>
</tr>
</table>
</div>
While an exact analytic solution to this eigenvalue problem is not (yet) known, [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] has determined that a good approximate solution is an eigenvector defined by the complex eigenfrequency,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\nu}{m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
~-1 ~\pm ~ i~\biggl[ \frac{3}{2(n+1)} \biggr]^{1/2} \beta \, ,
</math>
  </td>
</tr>
</table>
</div>
and, simultaneously, the complex eigenfunction,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Lambda</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- (4n+1)\beta^2 + (\beta\eta)^2 (n+1)^2[
2^3 \cos^2\theta - 3]  ~\pm~i~\beta [  2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(\beta\eta)^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x^2[1+x(3\cos\theta - \cos^3\theta )] \, .</math>
  </td>
</tr>
</table>
</div>
<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="6"><font size="+1">''Real'' Components of Various Terms</font></th>
</tr>
<tr>
  <td align="center">Order</td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM1}</math></td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM2}</math></td>
  <td align="center"><math>~f\ell^2\cdot \mathrm{TERM3}</math></td>
  <td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM4}</math></td>
  <td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM5}</math></td>
</tr>
<tr>
  <td align="center"><math>~\mathcal{O}(\beta^{-2})</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{n}{\beta^2}(1-2+1)</math></td>
</tr>
<tr>
  <td align="center"><math>~\mathcal{O}(\beta^{-1})</math></td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center">---</td>
  <td align="center"><math>~\frac{n}{\beta^2}(4-4)</math></td>
</tr>
<tr>
  <td align="center" rowspan="2"><math>~\mathcal{O}(\beta^0)</math></td>
  <td align="center"><math>~(n+1) [ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta ]f\ell^2 </math></td>
  <td align="center"><math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]f\ell^2 </math></td>
  <td align="center"><math>~-  2^2(n+1)^2f\ell^2</math> </td>
  <td align="center"><math>~-~n \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]</math></td>
  <td align="center"><math>~2^3 n (n+1)^2\biggl[ 4\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta-\frac{3}{2(n+1)} \biggr]</math></td>
</tr>
<tr>
  <td align="left" colspan="5">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Sigma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n+1)\biggl\{ \biggl[-6+2^4(n+1) - 2^4(n+1)\cos^2\theta
~-6 + 2^4(n+1)\cos^2\theta
-  2^2(n+1) \biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
-~n \biggl( \frac{x}{\beta}\biggr)^2 [2^5(n+1)\cos^2\theta]
+~12n \biggl( \frac{x}{\beta}\biggr)^2
+~2^5 n (n+1)\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta 
-~12 n \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(n+1)\biggl\{ \biggl[
-12 + 12(n+1)\biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
+~12n \biggl( \frac{x}{\beta}\biggr)^2 - 12n\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0 \, .</math> &nbsp; &nbsp; Amazing!
  </td>
</tr>
</table>
  </td>
</tr>
</table>
[[File:BetaErrorPlot02.png|right|350px|Beta Error Plot]]We have plugged this "Blaes85" approximate eigenvector into the five separate "TERM" expressions &#8212; analytically evaluating partial (1<sup>st</sup> and 2<sup>nd</sup>) derivatives along the way, as appropriate &#8212; then, with the aid of an Excel spreadsheet, have numerically evaluated each of the expressions over a range of coordinate locations <math>~(0 < x/\beta < 1; 0 \le \theta \le 2\pi)</math>.  The appropriate numerical sums of these TERMs are, indeed, nearly zero for slim <math>~(\beta \ll 1)</math> configurations. 
The log-log plot shown here, on the right, illustrates the behavior of the "TERM4 + TERM5" sum for the example parameter set, <math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>.  As the blue diamonds illustrate, the real part of this sum drops by approximately two orders of magnitude for every factor of ten drop in <math>~\beta</math>.  The total drop is roughly eight orders of magnitude over the displayed range, <math>~\beta = 1 ~\rightarrow~ 10^{-4}</math>.  As the salmon-colored squares in the same plot indicate, the imaginary part of the sum, "TERM4 + TERM5," is even closer to zero, dropping roughly 12 orders of magnitude over the same range of <math>~\beta</math>.  This indicates that, with the Blaes85 eigenvector, the real part of the sum of this pair of terms differs from zero by a residual whose leading-order term varies as <math>~\beta^{2}</math> while the corresponding imaginary part of the sum differs from zero by a residual whose leading-order term varies as <math>~\beta^{3}</math>.
As our [[#Imaginary_Parts|above analytic analysis]] shows, when each of the expressions for TERM4 and TERM5 is rewritten as a power series in <math>~\beta</math>, a sum of the two analytically specified TERMs results in precise cancellation of leading-order terms.  For the imaginary component of this sum, our derived expression for the residual is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathrm{Im}(\mathcal{R}_{45})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\mathrm{Im}[\mathrm{TERM4}+\mathrm{TERM5}]</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \beta^3 \biggl(\frac{x}{\beta}\biggr)^2 [2^7\cdot 3 (n+1)^3]^{1/2}[
3  + 6\cos^2\theta - 2\cos^4\theta ] + \mathcal{O}(\beta^4) \, .
</math>
  </td>
</tr>
</table>
</div>
The dotted, salmon-colored line of slope 3 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^3</math>-residual term.  It appears to precisely thread through the points (the salmon-colored squares) whose plot locations have been determined via our numerical spreadsheet evaluation of the imaginary component of the "TERM4 + TERM5" sum.  Additional confirmation that we have derived the correct analytic  expression for <math>~\mathrm{Im}(\mathcal{R}_{45})</math> comes from subtracting this analytically defined <math>~\beta^3</math> residual from the numerically determined sum:  The result is the green-dashed curve in the accompanying log-log plot, which appears to be a line of slope 4.
Analogously, for the real component of this sum, the precise expression for the residual is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathrm{Re}(\mathcal{R}_{45})</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\mathrm{Re}[\mathrm{TERM4}+\mathrm{TERM5}]</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] + \mathcal{O}(\beta^3) \, .
</math>
  </td>
</tr>
</table>
</div>
The dotted, light blue line of slope 2 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^2</math>-residual term.  It appears to precisely thread through the points (the light blue diamonds) whose plot locations have been determined via our numerical spreadsheet evaluation of the real part of the "TERM4 + TERM5" sum.  Notice that at the surface of the torus &#8212; that is, when <math>~x/\beta = 1</math> &#8212; this <math>~\beta^2</math>-residual goes to zero, in which case the leading order term in the "real" component residual will be drop to <math>~\mathcal{O}(\beta^3)</math>.


=See Also=
=See Also=

Latest revision as of 20:59, 24 May 2016

Stability Analyses of PP Tori (Part 2)

[Comment by J. E. Tohline on 24 May 2016]   This chapter contains a set of technical notes and accompanying discussion that I put together several months ago as I was trying to gain a foundational understanding of the results of a large study of instabilities in self-gravitating tori published by the Imamura & Hadley collaboration. I have come to appreciate that some of the logic and interpretation of published results that are presented, below, has serious flaws. Therefore, anyone reading this should be quite cautious in deciding what subsections provide useful insight. I have written a separate chapter titled, "Characteristics of Unstable Eigenvectors in Self-Gravitating Tori," that contains a much more trustworthy analysis of this very interesting problem.


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This is a direct extension of our Part 1 discussion. Here we continue our effort to check the validity of the Blaes85 eigenvector. The relevant reference is:

Start From Scratch

Basic Equations from Blaes85

  Blaes85

Eq. No.

<math>~(\beta\eta)^2</math>

<math>~=</math>

<math>~x^2(1+xb) \, ;</math>

(2.6)

<math>~b</math>

<math>~\equiv</math>

<math>~3\cos\theta - \cos^3\theta \, ;</math>

(2.6)

<math>~f</math>

<math>~=</math>

<math>~1-\eta^2 \, .</math>

(2.5)

  Blaes85

Eq. No.

<math>~LHS \equiv \hat{L}W</math>

<math>~=</math>

<math> ~fx^2 \cdot \frac{\partial^2 W}{\partial x^2} + f \cdot \frac{\partial^2 W}{\partial \theta^2} + \biggl[ \frac{fx(1-2x\cos\theta)}{(1-x\cos\theta)} + nx^2\cdot \frac{\partial f}{\partial x}\biggr]\frac{\partial W}{\partial x} </math>

 

 

 

<math> + \biggl[ \frac{fx\sin\theta}{(1-x\cos\theta)} + n\cdot \frac{\partial f}{\partial \theta}\biggr]\frac{\partial W}{\partial \theta} + \biggl[ \frac{2nx^2m^2}{\beta^2(1-x\cos\theta)^4} - \frac{m^2 x^2 f}{(1-x\cos\theta)^2} \biggr]W </math>

(4.2)

<math>~RHS</math>

<math>~=</math>

<math> ~-\frac{2nm^2}{\beta^2} \cdot (\beta\eta)^2 \biggl[ M \biggl(\frac{\nu}{m}\biggr)^2 + \frac{N}{m} \biggl(\frac{\nu}{m}\biggr)\biggr] W </math>

(4.1)

 

<math>~=</math>

<math> ~-\frac{2nm^2}{\beta^2} \biggl[ x^2 \biggl(\frac{\nu}{m}\biggr)^2 + \frac{2x^2}{(1-x\cos\theta)^2} \biggl(\frac{\nu}{m}\biggr)\biggr] W </math>

(4.2)

<math>~\frac{W}{A_{00}}</math>

<math>~=</math>

<math> ~1 + \beta^2 m^2 \biggl\{ 2\eta^2\cos^2\theta - \frac{3\eta^2}{4(n+1)} - \frac{(4n+1)}{4(n+1)^2} ~\pm~i~\biggl[ \frac{2^3\cdot 3}{(n+1)}\biggr]^{1/2} \eta\cos\theta \biggr\} </math>

(4.13)

<math>~\frac{\nu}{m}</math>

<math>~=</math>

<math> ~-1 ~\pm ~ i~\biggl[ \frac{3}{2(n+1)} \biggr]^{1/2} \beta </math>

(4.14)

Our Manipulation of These Equations

Analytic

<math>~\Lambda \equiv \frac{2^2(n+1)^2}{m^2}\biggl[\frac{W}{A_{00}}-1\biggr]</math>

<math>~=</math>

<math>~\beta^2 \biggl\{ 2^3(n+1)^2 \eta^2\cos^2\theta - 3\eta^2(n+1)^2 - (4n+1) ~\pm~i~[ 2^7\cdot 3(n+1)^3 ]^{1/2} \eta\cos\theta \biggr\} </math>

 

<math>~=</math>

<math>~- (4n+1)\beta^2 + (\beta\eta)^2 (n+1)^2[ 2^3 \cos^2\theta - 3] ~\pm~i~\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta \, ; </math>

<math>~\Rightarrow~~~~\frac{W}{A_{00}} </math>

<math>~=</math>

<math>~1+ \biggl[ \frac{m}{2(n+1)} \biggr]^2 \Lambda </math>


<math>~\frac{LHS}{A_{00}} </math>

<math>~=</math>

<math>~\biggl[ \frac{m}{2(n+1)} \biggr]^2 f ~\biggl[ x^2 \cdot \frac{\partial^2 \Lambda}{\partial x^2} + \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \biggl[ \frac{m}{2(n+1)} \biggr]^2\biggl[ \frac{fx(1-2x\cos\theta)}{(1-x\cos\theta)} + nx^2\cdot \frac{\partial f}{\partial x}\biggr]\frac{\partial \Lambda}{\partial x} </math>

 

 

<math> + \biggl[ \frac{m}{2(n+1)} \biggr]^2\biggl[ \frac{fx\sin\theta}{(1-x\cos\theta)} + n\cdot \frac{\partial f}{\partial \theta}\biggr]\frac{\partial \Lambda}{\partial \theta} + \biggl[ \frac{2nx^2m^2}{\beta^2(1-x\cos\theta)^4} - \frac{m^2 x^2 f}{(1-x\cos\theta)^2} \biggr]\biggl\{1+ \biggl[ \frac{m}{2(n+1)} \biggr]^2 \Lambda\biggr\} </math>

 

<math>~=</math>

<math>~\biggl[ \frac{m}{2(n+1)} \biggr]^2 f \biggl\{ ~\biggl[ x^2 \cdot \frac{\partial^2 \Lambda}{\partial x^2} + \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \biggl[ \frac{x(1-2x\cos\theta)}{(1-x\cos\theta)} \biggr]\frac{\partial \Lambda}{\partial x} + \biggl[ \frac{x\sin\theta}{(1-x\cos\theta)} \biggr]\frac{\partial \Lambda}{\partial \theta} - \biggl[ \frac{m^2 x^2 }{(1-x\cos\theta)^2} \biggr] \biggl[ \frac{2^2(n+1)^2}{m^2} + \Lambda\biggr]\biggr\} </math>

 

 

<math> + n\biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{ x^2\cdot \frac{\partial f}{\partial x}\cdot \frac{\partial \Lambda}{\partial x} ~+~ \frac{\partial f}{\partial \theta}\cdot \frac{\partial \Lambda}{\partial \theta} ~+~ \biggl[ \frac{2x^2m^2}{\beta^2(1-x\cos\theta)^4} \biggr]\biggl[ \frac{2^2(n+1)^2}{m^2} + \Lambda\biggr]\biggr\} </math>

 

<math>~=</math>

<math>~\frac{x^2 f}{(1-x\cos\theta)^2} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{ ~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] - [ 2^2(n+1)^2 + m^2\Lambda ]\biggr\} </math>

 

 

<math> + ~\frac{x^2 n}{\beta^2(1-x\cos\theta)^4} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl\{\beta^2 (1-x\cos\theta)^4\biggl[ \frac{\partial f}{\partial x}\cdot \frac{\partial \Lambda}{\partial x} ~+~ \frac{1}{x^2}\cdot \frac{\partial f}{\partial \theta}\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] ~+~ [ 2^3(n+1)^2 + 2m^2\Lambda ]\biggr\} \, . </math>

Also,

<math>~\frac{RHS}{A_{00}}</math>

<math>~=</math>

<math> ~-\frac{2n x^2}{\beta^2(1-x\cos\theta)^2} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl[ (1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)^2 + 2\biggl(\frac{\nu}{m}\biggr)\biggr] [ 2^2(n+1)^2 + m^2\Lambda ] </math>

 

<math>~=</math>

<math> ~-\frac{x^2n}{\beta^2(1-x\cos\theta)^4} \biggl[ \frac{m}{2(n+1)} \biggr]^2 \biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)\biggr] [ 2^3(n+1)^2 + 2m^2\Lambda ] \, . </math>

Putting the two together implies,

Definition of Eigenvalue Problem Associated with the Stability of Slim, Papaloizou-Pringle Tori

<math>~0</math>

<math>~=</math>

<math>~\frac{1}{x^2}\biggl[\frac{LHS}{A_{00}} - \frac{RHS}{A_{00}}\biggr]\biggl[ \frac{2(n+1)}{m} \biggr]^2 (1-x\cos\theta)^4</math>

 

<math>~=</math>

<math>~f (1-x\cos\theta)^2 \biggl\{ ~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] + \frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] - [ 2^2(n+1)^2 + m^2\Lambda ]\biggr\} </math>

 

 

<math> + ~\frac{n}{\beta^2} \biggl\{ (1-x\cos\theta)^4\biggl[ \frac{\partial \Lambda}{\partial x} \cdot \frac{\partial (\beta^2 f)}{\partial x} ~+~ \frac{\partial \Lambda}{\partial \theta} \cdot \frac{\partial (\beta^2 f/x^2)}{\partial \theta} \biggr] ~+~ \biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] [ 2^3(n+1)^2 + 2m^2\Lambda ] \biggr\} \, . </math>

The first line of this governing, two-line expression contains the function, <math>~f</math>, as a leading factor, while the leading factor in the second line is the ratio, <math>~n/\beta^2</math>. Presumably the three terms (hereafter, TERM1, TERM2, & TERM3, respectively) inside the curly brackets on the first line must cancel — to a sufficiently high order in <math>~x</math> — and, independently, the two terms (hereafter, TERM4 & Term5, respectively) inside the curly brackets on the second line must cancel. Furthermore, these cancellations must occur separately for the real parts and the imaginary parts of each bracketed expression.

Example Evaluation

Evaluating various terms using the parameter set,    <math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>    as begun in our "Part 1" analysis, we have:

TERM1

<math>~\equiv</math>

<math>~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] </math>

 

<math>~=</math>

<math>~ \biggl(\frac{7}{2^3} \biggr)^2\biggl\{ \frac{65}{2^3} + \frac{1}{2^4}\cdot [~4.269531250~] \biggr\} ~\pm~i~\biggl(\frac{7}{2^3} \biggr)^2\biggl\{ [~30.76957507~] + \frac{1}{2^4}\cdot (-1)[~5.773638858~] \biggr\}\beta </math>

 

<math>~=</math>

<math>~ \frac{7^2}{2^6} [ ~8.39184570 ~\pm~i~30.40872264~\beta] \, . </math>

TERM2

<math>~\equiv</math>

<math>~\frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] </math>

 

<math>~=</math>

<math>~ \frac{7}{2^5} [ ~-0.931640625 ~\pm~i~13.86780926~\beta] \, . </math>

TERM3

<math>~\equiv</math>

<math>~- [ 2^2(n+1)^2 + m^2\Lambda ] </math>

 

<math>~=</math>

<math>~ -\biggl\{~2^4 + m^2[~- 5\beta^2 + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr\}\, . </math>

The sum of these three terms gives,

TERM1 + TERM2 + TERM3

<math>~=</math>

<math>~ \frac{7^2}{2^6} [ ~8.39184570 ~\pm~i~30.40872264~\beta] +\frac{7}{2^5} [ ~-0.931640625 ~\pm~i~13.86780926~\beta] </math>

 

 

<math>~ -\biggl\{~2^4 + m^2[~- 5\beta^2 + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr\} </math>

 

<math>~=</math>

<math>~ 6.42500686 - 0.20379639 -~2^4 + 5m^2\beta^2 - m^2 0.167968750 </math>

 

 

<math>~ \pm~i~\biggl[23.28167827 + 3.03358328 - 8.031189202 ~m^2~\biggr]\beta </math>

 

<math>~=</math>

<math>~ -9.77878953+ 5m^2\beta^2 - m^2 0.167968750 ~ \pm~i~\biggl[26.31526155- 8.031189202 ~m^2\biggr]\beta </math>

Moving on to the last pair of terms …

TERM4

<math>~=</math>

<math>~ -x \ell^4\biggl[ (2+3xb)\cdot \frac{\partial\Lambda}{\partial x} - 3\sin^3\theta \cdot \frac{\partial\Lambda}{\partial \theta} \biggr] </math>

 

<math>~=</math>

<math>~ -x \ell^4\biggl[ (2+3xb)\cdot [~1.515625000~\pm~i~36.23373732 ~\beta] - 3\sin^3\theta \cdot [~-2.388335684~\pm~i~(-1)15.36617018 ~\beta] \biggr] </math>

 

<math>~=</math>

<math>~ -x\ell^4 [~9.248046874~\pm~i~139.7753772~\beta] </math>

TERM5 (Case B)

<math>~=</math>

<math>~ \biggl[ \ell^4 [1-0.75\beta^2~\pm~i~(-1)\sqrt{3}\beta] +2\ell^2[ -1~\pm~i~\sqrt{0.75}\beta ] + 1 \biggr] \cdot \biggl[~2^5 + 2\cancelto{1}{m^2}[~- 5\beta + 0.167968750~\pm~i~8.031189202 ~\beta]~\biggr] </math>

 

<math>~=</math>

<math>~ \biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr] \cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]~\pm~i~[(2)8.031189202 ~\beta]~\biggr] </math>

 

 

<math>~ \pm~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr] \cdot \biggl[~i~ [2^5 - 10\beta + (2)0.167968750]~-~[(2)8.031189202 ~\beta]~\biggr] </math>

 

<math>~=</math>

<math>~ \biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr] \cdot \biggl[~[2^5 - 10\beta + (2)0.167968750]\biggr] \pm~(-1)\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr] \cdot \biggl[[(2)8.031189202 ~\beta]~\biggr] </math>

 

 

<math>~ \pm~i~\biggl\{\biggl[ \ell^4 [1-0.75\beta^2] - 2\ell^2 + 1 \biggr] \cdot \biggl[[(2)8.031189202 ~\beta]~\biggr] +~\sqrt{3}\beta\biggl[ \ell^2-~\ell^4 \biggr] \cdot \biggl[~ [2^5 - 10\beta + (2)0.167968750]~\biggr] \biggr\} \, . </math>

Evaluating this TERM5 expression for the case of <math>~\beta = 1</math>, we have,


TERM5 (Case B)

<math>~=</math>

<math>~ \biggl[ 0.25\ell^4 - 2\ell^2 + 1 \biggr] \cdot \biggl[~[2^5 - 10 + (2)0.167968750]\biggr] \pm~(-1)\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr] \cdot \biggl[[(2)8.031189202 ]~\biggr] </math>

 

 

<math>~ \pm~i~\biggl\{\biggl[ \ell^4 [1-0.75] - 2\ell^2 + 1 \biggr] \cdot \biggl[[(2)8.031189202]~\biggr] +~\sqrt{3}\biggl[ \ell^2-~\ell^4 \biggr] \cdot \biggl[~ [2^5 - 10 + (2)0.167968750]~\biggr] \biggr\} </math>

 

<math>~=</math>

<math>~ [ -0.38470459 ] \cdot [22.3359375] \pm~(-1)[ ~0.31080502 ] \cdot [~16.0623784 ~] </math>

 

 

<math>~ \pm~i~\biggl\{[ -0.38470459 ] \cdot [~16.0623784 ~] +~[ ~0.31080502 ] \cdot [22.3359375]\biggr\} </math>

 

<math>~=</math>

<math>~[~-13.58500545~] \pm~i~[~0.76285080~] \, . </math>

Testing for Expected Cancellations

Note first that, adopting the shorthand notation,

<math>~\ell</math>

<math>~\equiv</math>

<math>~(1-x\cos\theta)</math>

<math>~\Rightarrow ~~~~\ell^2</math>

<math>~=</math>

<math>~1-2\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + \beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \, ;</math>

<math>~\ell^3</math>

<math>~=</math>

<math>~1-3\beta \biggl(\frac{x}{\beta}\biggr)\cos\theta + 3\beta^2 \biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta + \mathcal{O}(\beta^3) \, ;</math>

<math>~\ell^4</math>

<math>~=</math>

<math>~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 \, .</math>


Real Parts

TERM1

<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{\ell^2}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<math>~ - \cos^2\theta\biggl[232n + 241 \biggr] + 2^3\cdot 19(n+1)\cos^4\theta \biggr\} </math>

<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM1}}{(n+1)}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<math>~ \biggl[ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta\biggr] </math>

 

 

<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>

 

 

<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>

 

 

<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>

TERM2

<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{\ell}\biggr]</math>

<math>~=</math>

<math>~ -6(n+1) + 2^4(n+1)^2\cos^2\theta </math>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<math>~(n+1)\biggl[ -6 + 2^4(n+1)\cos^2\theta \biggr] </math>

 

 

<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>

 

 

<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>

<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM2}}{(n+1)}\biggr]</math>

<math>~=</math>

<math>~ \biggl[-6 + 2^4(n+1)\cos^2\theta \biggr]\biggl[1 - \beta\biggl(\frac{x}{\beta}\biggr)\cos\theta\biggr] </math>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<math>~ \biggl[-6 + 2^4(n+1)\cos^2\theta \biggr] </math>

 

 

<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>

 

 

<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>

 

 

<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>


Sum of TERM1 and TERM2

<math>~ \mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2}}{(n+1)} \biggr] </math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>


TERM3

<math>~\mathrm{Re}\biggl[\mathrm{TERM3}\biggr]</math>

<math>~=</math>

<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>

 

 

<math>~ - m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1)^2 b\biggl[2^3 \cos^2\theta - 3\biggr] </math>

<math>~\Rightarrow~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM3}}{(n+1)}\biggr]</math>

<math>~=</math>

<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>

 

 

<math>~ - m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr] \, . </math>

Sum of TERM1 + TERM2 + TERM3

Therefore,

<math>~ \mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}}{(n+1)} \biggr] </math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<math>~ - m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr] </math>

 

<math>~=</math>

<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>

TERM4

<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>

<math>~=</math>

<math>~ \biggl\{ (n+1)[2^3(n+1)\cos^2\theta -3]x(2+3xb)\biggr\} \cdot \biggl[ -x(2+3xb) \biggr] </math>

 

 

<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>

 

<math>~=</math>

<math>~ -~(n+1)[2^3(n+1)\cos^2\theta -3]x^2(2+3xb)^2 </math>

 

 

<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>

 

<math>~=</math>

<math>~ -~x^2 \cdot 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]\biggl(1+\frac{3xb}{2}\biggr)^2 </math>

 

 

<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>

 

<math>~=</math>

<math>~ -x\biggl\{~x[~18.37695315~] + x^2[~72.5625~] + x^3[~7.59375~]~~\biggr\} = -x[~9.24804688~]\, . </math>

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

<math>~\mathrm{Re}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

<math>~\Rightarrow ~~~ \mathrm{Re}\biggl[\mathrm{TERM4}\biggr]</math>

<math>~\approx</math>

<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>

 

 

<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> \, .

TERM5

Now, let's examine the TERM5 expressions.

<math>~\mathrm{Re}\biggl[\mathrm{TERM5}\biggr]</math>

<math>~=</math>

<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>

Case B:

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

Sum of TERM$ and TERM5

When added together, we obtain,

<math>~\mathrm{Re}[\mathrm{TERM4} + \mathrm{TERM5}]</math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<math>~ \beta^0 \cdot 2^3(n+1)^2\biggl\{1 - 2\ell^2 + \ell^4 \biggr\} </math>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<math>~ +~\beta^2 \biggl(\frac{x}{\beta}\biggr)^2 \cdot 2^5(n+1)^2 \cos^2\theta </math>

 

 

<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>

 

 

<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>

 

<math>~=</math>

<math>~-~\beta^2 2^2\cdot 3 (n+1) \biggl[1 - \biggl(\frac{x}{\beta}\biggr)^2\biggr] \, .</math>

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,


<math>~\Rightarrow ~~~ \mathrm{Re}\biggl[\mathrm{TERM4} + \mathrm{TERM5}\biggr]</math>

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<math>~ -\beta^22^2\cdot 3(n+1)\biggl[1-\biggl( \frac{x}{\beta}\biggr)^2 \biggr] \, . </math>

Exactly the same as the first time around.

Imaginary Parts

TERM1

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM1}}{\ell^2}\biggr]</math>

<math>~=</math>

<math>~ \beta\cos\theta [2^3\cdot 3(n+1)^3]^{1/2} \biggl[ \frac{b(4+3xb)}{(1+xb)^{3/2}} \biggr] </math>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

TERM2

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM2}}{\ell^2}\biggr]</math>

<math>~=</math>

<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>

 

 

<math>~- x^2 \cos\theta [ 9 +4\cos^2\theta -\cos^4\theta ] \biggr\}</math>

 

<math>~=</math>

<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>

TERM3

<math>~\mathrm{Im}\biggl[\mathrm{TERM3}\biggr]</math>

<math>~\equiv</math>

<math>~ -m^2\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta </math>

 

<math>~=</math>

<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>

TERM4

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<math>~ -x\biggl\{~[~109.8335164~] + x[~119.7674436~]~\biggr\}= -34.94384433 </math>

Alternatively we can write,

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\ell^4}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<math>~ -~ \frac{9b_0}{2} \cdot \beta^4 \biggl(\frac{x}{\beta}\biggr)^3 \sin^6\theta (1 + xb)^{-1/2} </math>


<math>~\Rightarrow ~~~ \mathrm{Im}\biggl[\frac{\mathrm{TERM4}}{\beta^2}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

<math>~</math>

<math>~=</math>

<math>~ \biggl\{ ~-27.45837910~-6.77631589 ~-0.70914934~ \biggr\}\times [~0.58618164~] =\biggl\{ ~-34.94384433~ \biggr\}\times [~0.58618164~] = -20.48343998 </math>

 

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

TERM5

<math>~\mathrm{Im}\biggl[\mathrm{TERM5}\biggr]</math>

<math>~=</math>

<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>

Case B:

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<math>~ \cancelto{1}{m^2} \biggl\{~-0.38470459~\biggr\} \cdot [ ~16.06237841~] +~[~0.31080502~] \cdot \biggl\{ 22.3359375\biggr\}= 0.76285080 \, . </math>


Let's rewrite both of these expressions in terms of a power series in <math>~\beta</math>.

<math>~\mathrm{Im}\biggl[\mathrm{TERM5}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

<math>~\Rightarrow~~~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

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

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM5}}{\beta^2}\biggr]</math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

Together

Together, then, we have:

<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM4}+\mathrm{TERM5}}{b_0\beta^2}\biggr]</math>

<math>~\approx</math>

<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>

 

<math>~\approx</math>

<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>

 

<math>~\approx</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>


Beta Error Plot

Work-in-progress.png

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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When added together, we obtain,

<math>~\mathrm{Im}[\mathrm{TERM4} + \mathrm{TERM5}]</math>

<math>~=</math>

<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>

 

 

<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>

 

 

<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>

 

 

<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>

 

 

<math>~ -~\beta [ 2^7\cdot 3 (n+1)^3]^{1/2} [\ell^2 -\ell^4] </math>

 

 

<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>

 

<math>~\approx</math>

<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>

 

 

<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>

 

 

<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>

 

<math>~\approx</math>

<math>~ -~\beta^1 [ 2^7\cdot 3 (n+1)^3]^{1/2} [1 - 1] </math>

 

 

<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>

 

 

<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>

Summary

As stated above, the eigenvalue problem that must be solved in order to identify the eigenfunction, <math>~\Lambda(x,\theta)</math>, and eigenfrequency, <math>~(\nu/m)</math>, of unstable (as well as stable) nonaxisymmetric modes in slim <math>~(\beta \ll 1)</math>, polytropic <math>~(n)</math> PP tori with uniform specific angular momentum is defined by the following two-dimensional <math>~(x,\theta)</math>, 2nd-order PDE:

<math>~0</math>

<math>~=</math>

<math>~f (1-x\cos\theta)^2 \biggl\{ ~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3} \biggr\} + ~\frac{n}{\beta^2} \biggl\{ \mathrm{TERM4} ~+~ \mathrm{TERM5}\biggr\} \, , </math>

where, <math>~f(x,\theta)</math> is the enthalpy distribution in the unperturbed, axisymmetric torus, and

<math>~\mathrm{TERM1}</math>

<math>~\equiv</math>

<math>~(1-x\cos\theta)^2\biggl[ \frac{\partial^2 \Lambda}{\partial x^2} + \frac{1}{x^2}\cdot \frac{\partial^2 \Lambda}{\partial \theta^2}\biggr] \, ,</math>

<math>~\mathrm{TERM2}</math>

<math>~\equiv</math>

<math>~\frac{(1-x\cos\theta)}{x} \biggl[ (1-2x\cos\theta) \frac{\partial \Lambda}{\partial x} + \sin\theta\cdot \frac{\partial \Lambda}{\partial \theta} \biggr] \, ,</math>

<math>~\mathrm{TERM3}</math>

<math>~\equiv</math>

<math>~- [ 2^2(n+1)^2 + m^2\Lambda ] \, ,</math>

<math>~\mathrm{TERM4}</math>

<math>~\equiv</math>

<math>~(1-x\cos\theta)^4\biggl[ \frac{\partial \Lambda}{\partial x} \cdot \frac{\partial (\beta^2 f)}{\partial x} ~+~ \frac{\partial \Lambda}{\partial \theta} \cdot \frac{\partial (\beta^2 f/x^2)}{\partial \theta} \biggr] \, ,</math>

<math>~\mathrm{TERM5}</math>

<math>~\equiv</math>

<math>~\biggl[ (1-x\cos\theta)^4\biggl(\frac{\nu}{m}\biggr)^2 + 2(1-x\cos\theta)^2\biggl(\frac{\nu}{m}\biggr)+ 1 \biggr] [ 2^3(n+1)^2 + 2m^2\Lambda ] \, .</math>

We also should appreciate that,

<math>~f\ell^2 \equiv f(1-x\cos\theta)^2</math>

<math>~=</math>

<math>~(1-\eta^2)(1-2x\cos\theta + x^2\cos^2\theta)</math>

 

<math>~=</math>

<math>~\biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 - \beta\biggl(\frac{x}{\beta}\biggr)^3 b\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>

 

<math>~=</math>

<math>~ \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr] \biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr] -~\beta\biggl(\frac{x}{\beta}\biggr)^3 b \biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr] </math>

 

<math>~=</math>

<math>~ \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr] \biggl[1-2\beta\biggl(\frac{x}{\beta}\biggr) \cos\theta + \beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggr] + \mathcal{O}(\beta^3) \, . </math>

If an exact solution, <math>~(\Lambda,\nu/m)</math>, to this eigenvalue problem were plugged into this governing PDE, we would expect that both of the following summations would be exactly zero at all meridional-plane <math>~(x,\theta)</math> locations throughout the torus:

<math>~0</math>

<math>~=</math>

<math>~\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3} \, ,</math>

<math>~0</math>

<math>~=</math>

<math>~\mathrm{TERM4} + \mathrm{TERM5} \, .</math>

While an exact analytic solution to this eigenvalue problem is not (yet) known, Blaes (1985) has determined that a good approximate solution is an eigenvector defined by the complex eigenfrequency,

<math>~\frac{\nu}{m}</math>

<math>~=</math>

<math> ~-1 ~\pm ~ i~\biggl[ \frac{3}{2(n+1)} \biggr]^{1/2} \beta \, , </math>

and, simultaneously, the complex eigenfunction,

<math>~\Lambda</math>

<math>~=</math>

<math>~- (4n+1)\beta^2 + (\beta\eta)^2 (n+1)^2[ 2^3 \cos^2\theta - 3] ~\pm~i~\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta \, , </math>

where,

<math>~(\beta\eta)^2</math>

<math>~=</math>

<math>~x^2[1+x(3\cos\theta - \cos^3\theta )] \, .</math>


Real Components of Various Terms
Order <math>~f\ell^2\cdot \mathrm{TERM1}</math> <math>~f\ell^2\cdot \mathrm{TERM2}</math> <math>~f\ell^2\cdot \mathrm{TERM3}</math> <math>~\frac{n}{\beta^2} \cdot\mathrm{TERM4}</math> <math>~\frac{n}{\beta^2} \cdot\mathrm{TERM5}</math>
<math>~\mathcal{O}(\beta^{-2})</math> --- --- --- --- <math>~\frac{n}{\beta^2}(1-2+1)</math>
<math>~\mathcal{O}(\beta^{-1})</math> --- --- --- --- <math>~\frac{n}{\beta^2}(4-4)</math>
<math>~\mathcal{O}(\beta^0)</math> <math>~(n+1) [ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta ]f\ell^2 </math> <math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]f\ell^2 </math> <math>~- 2^2(n+1)^2f\ell^2</math> <math>~-~n \biggl( \frac{x}{\beta}\biggr)^2 2^2 (n+1)[2^3(n+1)\cos^2\theta -3]</math> <math>~2^3 n (n+1)^2\biggl[ 4\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta-\frac{3}{2(n+1)} \biggr]</math>

<math>~\Sigma</math>

<math>~=</math>

<math>~ (n+1)\biggl\{ \biggl[-6+2^4(n+1) - 2^4(n+1)\cos^2\theta ~-6 + 2^4(n+1)\cos^2\theta - 2^2(n+1) \biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr] -~n \biggl( \frac{x}{\beta}\biggr)^2 [2^5(n+1)\cos^2\theta] +~12n \biggl( \frac{x}{\beta}\biggr)^2 +~2^5 n (n+1)\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta -~12 n \biggr\} </math>

 

<math>~=</math>

<math>~ (n+1)\biggl\{ \biggl[ -12 + 12(n+1)\biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr] +~12n \biggl( \frac{x}{\beta}\biggr)^2 - 12n\biggr\} </math>

 

<math>~=</math>

<math>~0 \, .</math>     Amazing!


Beta Error Plot

We have plugged this "Blaes85" approximate eigenvector into the five separate "TERM" expressions — analytically evaluating partial (1st and 2nd) derivatives along the way, as appropriate — then, with the aid of an Excel spreadsheet, have numerically evaluated each of the expressions over a range of coordinate locations <math>~(0 < x/\beta < 1; 0 \le \theta \le 2\pi)</math>. The appropriate numerical sums of these TERMs are, indeed, nearly zero for slim <math>~(\beta \ll 1)</math> configurations.


The log-log plot shown here, on the right, illustrates the behavior of the "TERM4 + TERM5" sum for the example parameter set, <math>~(n, \theta, x/\beta) = (1, \tfrac{\pi}{3}, \tfrac{1}{4})</math>. As the blue diamonds illustrate, the real part of this sum drops by approximately two orders of magnitude for every factor of ten drop in <math>~\beta</math>. The total drop is roughly eight orders of magnitude over the displayed range, <math>~\beta = 1 ~\rightarrow~ 10^{-4}</math>. As the salmon-colored squares in the same plot indicate, the imaginary part of the sum, "TERM4 + TERM5," is even closer to zero, dropping roughly 12 orders of magnitude over the same range of <math>~\beta</math>. This indicates that, with the Blaes85 eigenvector, the real part of the sum of this pair of terms differs from zero by a residual whose leading-order term varies as <math>~\beta^{2}</math> while the corresponding imaginary part of the sum differs from zero by a residual whose leading-order term varies as <math>~\beta^{3}</math>.


As our above analytic analysis shows, when each of the expressions for TERM4 and TERM5 is rewritten as a power series in <math>~\beta</math>, a sum of the two analytically specified TERMs results in precise cancellation of leading-order terms. For the imaginary component of this sum, our derived expression for the residual is,

<math>~\mathrm{Im}(\mathcal{R}_{45})</math>

<math>~\equiv</math>

<math>~\mathrm{Im}[\mathrm{TERM4}+\mathrm{TERM5}]</math>

 

<math>~=</math>

<math>~ - \beta^3 \biggl(\frac{x}{\beta}\biggr)^2 [2^7\cdot 3 (n+1)^3]^{1/2}[ 3 + 6\cos^2\theta - 2\cos^4\theta ] + \mathcal{O}(\beta^4) \, . </math>

The dotted, salmon-colored line of slope 3 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^3</math>-residual term. It appears to precisely thread through the points (the salmon-colored squares) whose plot locations have been determined via our numerical spreadsheet evaluation of the imaginary component of the "TERM4 + TERM5" sum. Additional confirmation that we have derived the correct analytic expression for <math>~\mathrm{Im}(\mathcal{R}_{45})</math> comes from subtracting this analytically defined <math>~\beta^3</math> residual from the numerically determined sum: The result is the green-dashed curve in the accompanying log-log plot, which appears to be a line of slope 4.


Analogously, for the real component of this sum, the precise expression for the residual is,

<math>~\mathrm{Re}(\mathcal{R}_{45})</math>

<math>~\equiv</math>

<math>~\mathrm{Re}[\mathrm{TERM4}+\mathrm{TERM5}]</math>

 

<math>~=</math>

<math>~ -\beta^22^2\cdot 3(n+1)\biggl[1-\biggl( \frac{x}{\beta}\biggr)^2 \biggr] + \mathcal{O}(\beta^3) \, . </math>

The dotted, light blue line of slope 2 that has been drawn in our accompanying log-log plot was generated using this analytic expression for the <math>~\beta^2</math>-residual term. It appears to precisely thread through the points (the light blue diamonds) whose plot locations have been determined via our numerical spreadsheet evaluation of the real part of the "TERM4 + TERM5" sum. Notice that at the surface of the torus — that is, when <math>~x/\beta = 1</math> — this <math>~\beta^2</math>-residual goes to zero, in which case the leading order term in the "real" component residual will be drop to <math>~\mathcal{O}(\beta^3)</math>.

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation