User:Tohline/Appendix/Ramblings/PPToriPt2 - Revision history

Tohline: Type in cautionary introductory paragraph

2016-05-24T20:59:29Z

Type in cautionary introductory paragraph

← Older revision		Revision as of 20:59, 24 May 2016
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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>   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, "[[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.

			{{LSU_WorkInProgress}}


	{{LSU_HBook_header}}		{{LSU_HBook_header}}

Tohline: /* TERM1 */ Working on power-series expression for imaginary component of TERM1

2016-05-03T01:01:59Z

TERM1: Working on power-series expression for imaginary component of TERM1

← Older revision		Revision as of 01:01, 3 May 2016
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\beta\~~cos~~\~~theta [2^3~~\~~cdot 3~~(~~n+1~~)^3]^{1~~/2}~~ \biggl[ \frac{~~b(4+3xb)~~}{~~(1+xb~~)^{3/2}~~} \biggr]~~		\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>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	+\frac{1}{x^2} \~~cdot~~		-\frac{\beta b_0}{2^2x\cos\theta} \biggl[ 1 +\beta \biggl(\frac{x}{\beta}\biggr)b \biggr]^{1/2}\biggl\{
	(-1)\beta ~~[2^7~~\~~cdot 3~~ (~~n+1~~)^3 ]^{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}
	~~(\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}		+ 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\}		\biggr\}
	</math>		</math>

Tohline: /* Imaginary Parts */

2016-05-03T00:45:09Z

Imaginary Parts

← Older revision		Revision as of 00:45, 3 May 2016
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	<td align="right">		<td align="right">
	<math>~\mathrm{Im}\biggl[\frac{\mathrm{TERM1}}{\ell^2}\biggr]</math>		<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">

			</td>
			<td align="center">

			</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">

	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* TERM2 */ Worked on series expression for imaginary component of TERM2

2016-05-03T00:40:45Z

TERM2: Worked on series expression for imaginary component of TERM2

← Older revision		Revision as of 00:40, 3 May 2016
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	<math>~- x^2 \cos\theta [ 9 +4\cos^2\theta -\cos^4\theta ]		<math>~- x^2 \cos\theta [ 9 +4\cos^2\theta -\cos^4\theta ]
	\biggr\}</math>		\biggr\}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

Tohline: /* TERM3 */ Work on form of imaginary part of TERM3

2016-05-03T00:26:09Z

TERM3: Work on form of imaginary part of TERM3

← Older revision		Revision as of 00:26, 3 May 2016
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	<math>~		<math>~
	-m^2\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta		-m^2\beta [ 2^7\cdot 3(n+1)^3 ]^{1/2} (\beta\eta) \cos\theta
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</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>		</math>
	</td>		</td>
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	</table>		</table>
	</div>		</div>


	=====TERM4=====		=====TERM4=====

Tohline: /* Summary */ Finished showing cancellation (to order beta^0) of five terms in real component of the Blaes85 eigenvector solution

2016-05-02T16:11:56Z

Summary: Finished showing cancellation (to order beta^0) of five terms in real component of the Blaes85 eigenvector solution

← Older revision		Revision as of 16:11, 2 May 2016
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	<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]		<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>		[ 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">

			</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">

			</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">

			</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>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="center" rowspan="2"><math>~\mathcal{O}(\beta^0)</math></td>		<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 ] </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 ]</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)^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>~-~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>		<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>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	(n+1)\biggl\{ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta		(n+1)\biggl\{ \biggl[-6+2^4(n+1) - 2^4(n+1)\cos^2\theta
	~-6 + 2^4(n+1)\cos^2\theta		~-6 + 2^4(n+1)\cos^2\theta
	- 2^2(n+1)		- 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]		-~n \biggl( \frac{x}{\beta}\biggr)^2 [2^5(n+1)\cos^2\theta]
	+~12n \biggl( \frac{x}{\beta}\biggr)^2		+~12n \biggl( \frac{x}{\beta}\biggr)^2
Line 4,238:		Line 4,301:
	<td align="left">		<td align="left">
	<math>~		<math>~
	(n+1)\biggl\{ -12+~~2^4~~(n+1)		(n+1)\biggl\{ \biggl[
	- 2^2~~(n+1)~~		-12 + 12(n+1)\biggr]\cdot \biggl[ 1-\biggl(\frac{x}{\beta}\biggr)^2 \biggr]
	+~12n \biggl( \frac{x}{\beta}\biggr)^2		+~12n \biggl( \frac{x}{\beta}\biggr)^2 - 12n\biggr\}
	-~~~12 n~~ \biggr\}
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 \, .</math>     Amazing!
	</td>		</td>
	</tr>		</tr>

Tohline: /* Example Evaluation */

2016-05-02T15:46:17Z

Example Evaluation

← Older revision		Revision as of 15:46, 2 May 2016
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	~ \pm~i~\biggl[26.31526155- 8.031189202 ~m^2\biggr]\beta		~ \pm~i~\biggl[26.31526155- 8.031189202 ~m^2\biggr]\beta
	</math>		</math>
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	~~We also need to know that,~~
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

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

Tohline: /* Summary */ Attempt to simplify summation expression

2016-05-02T15:41:12Z

Summary: Attempt to simplify summation expression

← Older revision		Revision as of 15:41, 2 May 2016
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	<table border="1" cellpadding="5" align="center">		<table border="1" cellpadding="5" align="center">
	<tr>		<tr>
	<th align="center" colspan="7"><font size="+1">''Real'' Components of Various Terms</font></th>		<th align="center" colspan="6"><font size="+1">''Real'' Components of Various Terms</font></th>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM4}</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>		<td align="center"><math>~\frac{n}{\beta^2} \cdot\mathrm{TERM5}</math></td>
	~~<td align="center"><math>~\Sigma</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center">---</td>		<td align="center">---</td>
	<td align="center"><math>~\frac{n}{\beta^2}(1-2+1)</math></td>		<td align="center"><math>~\frac{n}{\beta^2}(1-2+1)</math></td>
	~~<td align="center"><math>~0</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
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	<td align="center">---</td>		<td align="center">---</td>
	<td align="center"><math>~\frac{n}{\beta^2}(4-4)</math></td>		<td align="center"><math>~\frac{n}{\beta^2}(4-4)</math></td>
	~~<td align="center"><math>~0</math></td>~~
	</tr>		</tr>
	<tr>		<tr>
	<td align="center"><math>~\mathcal{O}(\beta^0)</math></td>		<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 ] </math></td>		<td align="center"><math>~(n+1) [ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta ] </math></td>
	<td align="center"><math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]</math></td>		<td align="center"><math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]</math></td>
	<td align="center"><math>~- 2^2(n+1)^2</math></td>		<td align="center"><math>~- 2^2(n+1)^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>~-~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>		<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>
	<td align="center"><math>~</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\{ -6+2^4(n+1) - 2^4(n+1)\cos^2\theta
			~-6 + 2^4(n+1)\cos^2\theta
			- 2^2(n+1)
			-~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">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(n+1)\biggl\{ -12+2^4(n+1)
			- 2^2(n+1)
			+~12n \biggl( \frac{x}{\beta}\biggr)^2
			-~12 n \biggr\}
			</math>
			</td>
			</tr>
			</table>
			</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Summary */ Begin filling out table of various "order" expressions

2016-05-02T15:05:37Z

Summary: Begin filling out table of various "order" expressions

← Older revision		Revision as of 15:05, 2 May 2016
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	</table>		</table>
	</div>		</div>


			<table border="1" cellpadding="5" align="center">
			<tr>
			<th align="center" colspan="7"><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>
			<td align="center"><math>~\Sigma</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>
			<td align="center"><math>~0</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>
			<td align="center"><math>~0</math></td>
			</tr>
			<tr>
			<td align="center"><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 ] </math></td>
			<td align="center"><math>~(n+1) [-6 + 2^4(n+1)\cos^2\theta ]</math></td>
			<td align="center"><math>~- 2^2(n+1)^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>
			<td align="center"><math>~</math></td>
			</tr>
			</table>


	[[File:BetaErrorPlot02.png\|right\|350px\|Beta Error Plot]]We have plugged this "Blaes85" approximate eigenvector into the five separate "TERM" expressions — analytically evaluating partial (1<sup>st</sup> and 2<sup>nd</sup>) 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.		[[File:BetaErrorPlot02.png\|right\|350px\|Beta Error Plot]]We have plugged this "Blaes85" approximate eigenvector into the five separate "TERM" expressions — analytically evaluating partial (1<sup>st</sup> and 2<sup>nd</sup>) 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.

Tohline: /* Sum of TERM1 + TERM2 + TERM3 */

2016-05-02T01:22:01Z

Sum of TERM1 + TERM2 + TERM3

← Older revision		Revision as of 01:22, 2 May 2016
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	<math>~~~\Rightarrow~~~~~~\mathrm{Re}\biggl[\frac{\mathrm{TERM3}}{(n+1)}\biggr]</math>		<math>~
			\mathrm{Re}\biggl[ \frac{\mathrm{TERM1} + \mathrm{TERM2} + \mathrm{TERM3}}{(n+1)} \biggr]
			</math>
	</td>		</td>
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	<math>~- 2^2(n+1) +		<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]		\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>
	</td>		</td>
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	<math>~		<math>~
	~~- m^2~~ \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\~~biggl[~~2^3 \cos^2\theta ~~- 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>
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			</td>
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	<math>~		<math>~
	\~~mathrm~~{Re}\biggl[ \~~frac{~~\~~mathrm{TERM1}~~ + \~~mathrm{TERM2}}{~~(n+1)} \biggr]		+ \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>
			</td>
			</tr>

			<tr>
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	</td>		</td>
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	~~<math>~=</math>~~
	</td>		</td>
	<td align="left">		<td align="left">
	<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\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>
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	<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\}		+ 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>
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	<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\}		-\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>
	</td>		</td>
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	<math>~		<math>~
	~~- 2~~\beta^2\biggl(\frac{x}{\beta}\biggr)^2\cos^2\theta \biggl\{(~~136n~~ +~~112~~)		+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(16n +13)
	- \cos^2\theta\biggl[~~224n~~ + ~~233~~ \biggr] + 2^3\cdot 19(n+1)\cos^4\theta		- \cos^2\theta\biggl[232n + 241 \biggr] + 2^3\cdot 19(n+1)\cos^4\theta
	\biggr\}		\biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

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	<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		- m^2 \beta^3\biggl(\frac{x}{\beta}\biggr)^3 (n+1) b\biggl[2^3 \cos^2\theta - 3\biggr]
	\biggr\}
	</math>		</math>
	</td>		</td>
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	~~ ~~		<math>~=</math>
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	<math>~		<math>~12n
	+\beta^3\biggl(\frac{x}{\beta}\biggr)^3 \cos^3\theta \biggl\{ 3^2(~~16n~~ +13)		+ \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)
	- \cos^2\theta~~\biggl[232n + 241~~ ~~\biggr]~~ + 2^~~3\cdot 19~~(n+1)\cos^4\theta
	\biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>

	</table>		</table>
	</div>		</div>



	=====TERM4=====		=====TERM4=====