User:Tohline/Apps/Wong1973Potential - Revision history

Tohline: /* Summary */

2018-10-04T18:41:07Z

Summary

← Older revision		Revision as of 18:41, 4 October 2018
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	<math>~		<math>~
	- \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{		- \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{
	2\biggl[1+\frac{3}{2^4}~\epsilon^2 \biggr] (1 + \epsilon^2)^{3 / 2} (1-\epsilon^2)^{-1}		\biggl[2+\frac{3}{2^3}~\epsilon^2 \biggr] (1 + \epsilon^2)^{3 / 2} (1-\epsilon^2)^{-1}
	-~\frac{2^{4}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z)~\cos(2\theta)		-~\frac{2^{4}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z)~\cos(2\theta)
	\biggr\}		\biggr\}
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	<math>~		<math>~
	- \biggl[ \frac{E(k)}{R} \biggr] \biggl\{		- \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
	~ \frac{\epsilon^~~4}{~~2}~~\biggl[ \frac~~{~~\pi^2~~}~~{2^2~~(1-\epsilon^2)~~} \biggr] \frac{2~~^~~{4}}~~{3~~\pi^2} \biggl[ \epsilon^{-~~2} (1 + \epsilon^2)^{~~3 / 2~~} ~~\biggr]~~ \cos\theta		~ \frac{2\epsilon^2}{3} (1 + \epsilon^2)^{3 / 2} (1-\epsilon^2)^{-1} \cos\theta
	~+~ ~~C_2(z) \cdot~~ \frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \cos(\theta)		~+~\frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z) \biggl[ \cos(\theta)
	~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)		~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)
	~+~\cos (3\theta ) \biggr]		~+~\cos (3\theta ) \biggr]
	\biggr\}		\biggr\} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* Summary */

2018-10-04T18:34:07Z

Summary

← Older revision		Revision as of 18:34, 4 October 2018
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	+~\biggl( \frac{2}{3} \biggr)		+~\biggl( \frac{2}{3} \biggr)
	C_2(z)~\cos (3\theta )		C_2(z)~\cos (3\theta )
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{
			\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr] 3\epsilon^2 \biggl[1+\frac{3}{2^4}~\epsilon^2 \biggr]\frac{2^{3}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr]
			-~\frac{2^{4}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z)~\cos(2\theta)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			- \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
			~ \frac{\epsilon^4}{2}\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr] \frac{2^{4}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \cos\theta
			~+~ C_2(z) \cdot \frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \cos(\theta)
			~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)
			~+~\cos (3\theta ) \biggr]
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{
			2\biggl[1+\frac{3}{2^4}~\epsilon^2 \biggr] (1 + \epsilon^2)^{3 / 2} (1-\epsilon^2)^{-1}
			-~\frac{2^{4}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] C_2(z)~\cos(2\theta)
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			- \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
			~ \frac{\epsilon^4}{2}\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr] \frac{2^{4}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \cos\theta
			~+~ C_2(z) \cdot \frac{2^{5}}{3^2\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \cos(\theta)
			~+~\biggl( \frac{4c^2 }{RR_1} \biggr)~\cos(2\theta)
			~+~\cos (3\theta ) \biggr]
	\biggr\}		\biggr\}
	</math>		</math>

Tohline: /* Summary */

2018-10-04T16:56:59Z

Summary

← Older revision		Revision as of 16:56, 4 October 2018
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- 2^{3 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ ~~C_0~~(z)		- 2^{3 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{
	-~\biggl( \frac{2}{3} \biggr)		\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr] 3\epsilon^2 \biggl[1+\frac{3}{2^4}~\epsilon^2 \biggr]
	C_2(z)~\cos(2\theta) \biggr\}		-~\biggl( \frac{2}{3} \biggr) C_2(z)~\cos(2\theta)
			\biggr\}
	</math>		</math>
	</td>		</td>
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	<math>~		<math>~
	- 2^{5 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{E(k)}{R} \biggr] \biggl\{		- 2^{5 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
	~ \biggl[ ~~C_1~~(z) + \frac{2}{3} \cdot C_2(z) ~~\biggr]~~\cos(\theta)		~ \frac{\epsilon^4}{2}\biggl[ \frac{\pi^2}{2^2(1-\epsilon^2)} \biggr]\cos\theta + \frac{2}{3} \cdot C_2(z) \cos(\theta)
	+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)		+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
	C_2(z)~\cos(2\theta)		C_2(z)~\cos(2\theta)

Tohline: /* Summary */

2018-10-04T16:46:49Z

Summary

← Older revision		Revision as of 16:46, 4 October 2018
Line 6,434:		Line 6,434:
	<tr>		<tr>
	<td align="right">		<td align="right">
	~~<math>~\frac{\pi}{a}\biggl[\Phi_{\mathrm{W}0} + \Phi_{\mathrm{W}1} + \Phi_{\mathrm{W}2}\biggr]_\mathrm{exterior} </math>~~
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Summary */

2018-10-04T16:39:59Z

Summary

← Older revision		Revision as of 16:39, 4 October 2018
Line 6,371:		Line 6,371:

	See, also, a separate [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|table giving example evaluations of elliptic integrals]], and other [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|useful elliptic integral expressions/relations]].		See, also, a separate [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|table giving example evaluations of elliptic integrals]], and other [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|useful elliptic integral expressions/relations]].
			</td>
			</tr>
			</table>


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

			<tr>
			<td align="right">
			<math>~D_0 \equiv \frac{2^{3 / 2}}{3\pi^2} \biggl[\frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(~~\cosh\eta_0~~)		- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(z)
	-~\biggl( \frac{2}{3} \biggr)		-~\biggl( \frac{2}{3} \biggr)
	C_2(~~\cosh\eta_0~~)~\cos(2\theta) \biggr\}		C_2(z)~\cos(2\theta) \biggr\}
	</math>		</math>
	</td>		</td>
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	<math>~		<math>~
	- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R} \biggr] \biggl\{		- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
	~ \biggl[ C_1(\~~cosh~~\~~eta_0~~) + \frac{2}{3} \cdot C_2(\~~cosh~~\~~eta_0~~) \biggr]\cos(\theta)		~ \biggl[ C_1(z) + \frac{2}{3} \cdot C_2(z) \biggr]\cos(\theta)
			+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
			C_2(z)~\cos(2\theta)
			+~\biggl( \frac{2}{3} \biggr)
			C_2(z)~\cos (3\theta )
			\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{\pi}{a}\biggl[\Phi_{\mathrm{W}0} + \Phi_{\mathrm{W}1} + \Phi_{\mathrm{W}2}\biggr]_\mathrm{exterior} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- 2^{3 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(z)
			-~\biggl( \frac{2}{3} \biggr)
			C_2(z)~\cos(2\theta) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			- 2^{5 / 2} \cdot \frac{2^{3 / 2}}{3\pi^2} \biggl[ \epsilon^{-2} (1 + \epsilon^2)^{3 / 2} \biggr] \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
			~ \biggl[ C_1(z) + \frac{2}{3} \cdot C_2(z) \biggr]\cos(\theta)
	+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)		+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
	C_2(~~\cosh\eta_0~~)~\cos(2\theta)		C_2(z)~\cos(2\theta)
	+~\biggl( \frac{2}{3} \biggr)		+~\biggl( \frac{2}{3} \biggr)
	C_2(~~\cosh\eta_0~~)~\cos (3\theta )		C_2(z)~\cos (3\theta )
	\biggr\}		\biggr\}
	</math>		</math>

Tohline: /* Summary */

2018-10-04T16:33:45Z

Summary

← Older revision		Revision as of 16:33, 4 October 2018
Line 6,375:		Line 6,375:
	</table>		</table>

	From above, when added together, the three leading terms in Wong's expression for the potential give,
			From [[#ThreeTermsAdded\|above]], when added together, the three leading terms in Wong's expression for the exterior potential give,

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

Tohline: /* Focus on Exterior Solution */

2018-10-04T16:32:04Z

Focus on Exterior Solution

← Older revision		Revision as of 16:32, 4 October 2018
Line 3,043:		Line 3,043:
	= \biggl[ \frac{2}{\coth\eta+1}\biggr]^{1 / 2} = \sqrt{1-e^{-2\eta}} \, ,</math>		= \biggl[ \frac{2}{\coth\eta+1}\biggr]^{1 / 2} = \sqrt{1-e^{-2\eta}} \, ,</math>
	</div>		</div>
	these three terms added together give,		<span id="ThreeTermsAdded">these three terms added together give,</span>

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

Tohline: /* Summary */

2018-10-04T16:30:44Z

Summary

← Older revision		Revision as of 16:30, 4 October 2018
Line 6,371:		Line 6,371:

	See, also, a separate [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|table giving example evaluations of elliptic integrals]], and other [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|useful elliptic integral expressions/relations]].		See, also, a separate [[User:Tohline/2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|table giving example evaluations of elliptic integrals]], and other [[User:Tohline/2DStructure/ToroidalGreenFunction#Appendix_B:_Elliptic_Integrals\|useful elliptic integral expressions/relations]].
			</td>
			</tr>
			</table>

			From above, when added together, the three leading terms in Wong's expression for the potential give,

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

			<tr>
			<td align="right">
			<math>~\frac{\pi}{a}\biggl[\Phi_{\mathrm{W}0} + \Phi_{\mathrm{W}1} + \Phi_{\mathrm{W}2}\biggr]_\mathrm{exterior} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- 2^{3 / 2} D_0 \biggl[ \frac{2K(\mu)}{R_1 + R} \biggr] \biggl\{ C_0(\cosh\eta_0)
			-~\biggl( \frac{2}{3} \biggr)
			C_2(\cosh\eta_0)~\cos(2\theta) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			- 2^{5 / 2}D_0 \biggl[ \frac{E(k)}{R} \biggr] \biggl\{
			~ \biggl[ C_1(\cosh\eta_0) + \frac{2}{3} \cdot C_2(\cosh\eta_0) \biggr]\cos(\theta)
			+~\biggl( \frac{2}{3} \biggr)\biggl( \frac{4c^2 }{RR_1} \biggr)
			C_2(\cosh\eta_0)~\cos(2\theta)
			+~\biggl( \frac{2}{3} \biggr)
			C_2(\cosh\eta_0)~\cos (3\theta )
			\biggr\}
			</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Phase 1C */

2018-10-04T16:23:55Z

Phase 1C

← Older revision		Revision as of 16:23, 4 October 2018
Line 6,080:		Line 6,080:
	~+~ \epsilon^3 (2 - 4\epsilon +6\epsilon^2)		~+~ \epsilon^3 (2 - 4\epsilon +6\epsilon^2)
	~-~ \frac{\epsilon^4}{2^2} (1 - 2\epsilon +3\epsilon^2)		~-~ \frac{\epsilon^4}{2^2} (1 - 2\epsilon +3\epsilon^2)
			~+~ 11~\cdot \frac{\epsilon^4}{2^2}
			~+~ \mathcal{O}(\epsilon^{5})
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			(- 4\epsilon^4 )
			~+~ (6\epsilon^4 )
			~-~ 4\epsilon^4
			~-~ \frac{\epsilon^4}{2^2}
	~+~ 11~\cdot \frac{\epsilon^4}{2^2}		~+~ 11~\cdot \frac{\epsilon^4}{2^2}
	~+~ \mathcal{O}(\epsilon^{5})		~+~ \mathcal{O}(\epsilon^{5})
Line 6,128:		Line 6,147:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{\epsilon^2}{3} (1+\~~epsilon^2)^~~{~~3 / 2~~} (~~1 -~~ \epsilon~~^2 )~~^{-1} , .		\frac{\epsilon^2}{3} ~+~ \mathcal{O}(\epsilon^{3})\, .
	</math>		</math>
	</td>		</td>

Tohline: /* Phase 1C */

2018-10-04T16:13:33Z

Phase 1C

← Older revision		Revision as of 16:13, 4 October 2018
Line 5,991:		Line 5,991:
	</table>		</table>
	and, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial\|binomial series expansion]], we can write,		and, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial\|binomial series expansion]], we can write,


	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 6,098:		Line 6,097:
	\frac{\epsilon^4}{2}		\frac{\epsilon^4}{2}
	~+~ \mathcal{O}(\epsilon^{5}) \, .		~+~ \mathcal{O}(\epsilon^{5}) \, .
			</math>
			</td>
			</tr>
			</table>

			Therefore,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~2^{3/2}D_0 \cdot C_1</math>
			</td>
			<td align="center">
			<math>~\approx</math>
			</td>
			<td align="left">
			<math>~
			\frac{2^3}{3\pi^2} \biggl[\epsilon^{-2}(1+\epsilon^2)^{3 / 2}\biggr] \biggl[ \frac{\pi^2}{2^2(1 - \epsilon^2 )} \biggr] \frac{\epsilon^4}{2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\epsilon^2}{3} (1+\epsilon^2)^{3 / 2} (1 - \epsilon^2 )^{-1} , .
	</math>		</math>
	</td>		</td>