User:Tohline/Appendix/Mathematics/ToroidalFunctions - Revision history

Tohline: /* Summary of Toroidal Coordinates and Toroidal Functions */

2018-07-01T17:41:09Z

Summary of Toroidal Coordinates and Toroidal Functions

← Older revision		Revision as of 17:41, 1 July 2018
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	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	See [[#Caption\|relevant caption]].		See [[User:Tohline/Appendix/Equation_templates#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="left" colspan="3">		<td align="left" colspan="3">
	See [[#Caption\|relevant caption]].		See [[User:Tohline/Appendix/Equation_templates#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>
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	</tr>		</tr>
	</table>		</table>

	~~<div align="center" id="Caption">~~
	~~<table border="1" cellpadding="8" width="85%" align="center">~~
	~~<tr>~~
	~~<td align="left">~~
	'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.

	For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of the 1971 (19<sup>th</sup>) edition of the CRC's ''Standard Mathematical Tables'', published by the Chemical Rubber Co., Cleveland, Ohio, U.S.A..

	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.

	ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz & Stegun (1995)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>, which is entirely consistent with the lower-precision value that we have extracted from [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	===Asymptotic Behavior===		===Asymptotic Behavior===

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T22:17:17Z

Wong Toroidal Coordinates

← Older revision		Revision as of 22:17, 28 June 2018
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	==Wong Toroidal Coordinates==		==Wong Toroidal Coordinates==

			===Wong's Expression for the Potential===
	This chapter has been put together in an effort to lay the groundwork for an evaluation of [[User:Tohline/Apps/DysonWongTori#Wong_.281973.2C_1974.29\|Wong's (1973)]] derived expression for the gravitational potential both inside and outside of a uniform-density, axisymmetric torus. After multiplying his expression by the negative of {{ User:Tohline/Math/C_GravitationalConstant }}, then replacing his total charge, <math>~q</math>, with the total mass, <math>~M</math>, Wong's ''interior'' (i.e., <math>~\eta^' > \eta_0</math>) solution is,		This chapter has been put together in an effort to lay the groundwork for an evaluation of [[User:Tohline/Apps/DysonWongTori#Wong_.281973.2C_1974.29\|Wong's (1973)]] derived expression for the gravitational potential both inside and outside of a uniform-density, axisymmetric torus. After multiplying his expression by the negative of {{ User:Tohline/Math/C_GravitationalConstant }}, then replacing his total charge, <math>~q</math>, with the total mass, <math>~M</math>, Wong's ''interior'' (i.e., <math>~\eta^' > \eta_0</math>) solution is,

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

			===Summary of Toroidal Coordinates and Toroidal Functions===

	<table border="1" cellpadding="8" width="85%" align="center">		<table border="1" cellpadding="8" width="85%" align="center">
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	</div>		</div>

	Asymptotic ~~behavior:~~		===Asymptotic Behavior===

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

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	</table>		</table>

			===Dimensionless Potential Expression===

	Given that the portion of the leading term in Wong's expression that sits inside the square brackets is equivalent to the density, <math>~\rho_0 = M/V</math>, of the torus material — that is, given that,		Given that the portion of the leading term in Wong's expression that sits inside the square brackets is equivalent to the density, <math>~\rho_0 = M/V</math>, of the torus material — that is, given that,

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T20:23:24Z

Wong Toroidal Coordinates

← Older revision		Revision as of 20:23, 28 June 2018
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	<math>~		<math>~
	\frac{\Gamma(\frac{1}{2}) \sqrt{\pi} }{ \Gamma(1) (2\chi)^{\frac{1}{2}} }		\frac{\Gamma(\frac{1}{2}) \sqrt{\pi} }{ \Gamma(1) (2\chi)^{\frac{1}{2}} }
	= \frac{\pi}{ (2\chi)^~~{\frac~~{1}{2}} } \, .		= \frac{\pi}{ (2\chi)^{1 / 2} } \, .
	</math>		</math>
	</td>		</td>
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	<td align="center">		<td align="center">
	<math>~		<math>~
	\frac{\pi }{2^~~{\frac~~{5}{2}} \chi^~~{\frac~~{3}{2}} }		\frac{\pi }{2^{5/2} \chi^{3/2} }
	</math>		</math>
	</td>		</td>
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	<math>~		<math>~
	\frac{\pi }{2^{\frac{9}{2}} \chi^{\frac{5}{2}} } \cdot \biggl[ \frac{(3)!! }{ 2 }\biggr]		\frac{\pi }{2^{\frac{9}{2}} \chi^{\frac{5}{2}} } \cdot \biggl[ \frac{(3)!! }{ 2 }\biggr]
			=
			\frac{360~\pi }{2^{9/2} \chi^{5/2} }
			=
			\frac{45~\pi }{2^{3/2} \chi^{5/2} }
	</math>		</math>
	</td>		</td>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T20:10:32Z

Wong Toroidal Coordinates

← Older revision		Revision as of 20:10, 28 June 2018
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	</table>		</table>
	</div>		</div>

			Asymptotic behavior:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\lim_{\chi\rightarrow \infty} Q_{m-\frac{1}{2}}(\chi)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\Gamma(m+\frac{1}{2}) \sqrt{\pi} }{ \Gamma(m+1) (2\chi)^{m+\frac{1}{2}} }
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\sqrt{\pi} }{ (2\chi)^{m+\frac{1}{2}} } \cdot \biggl[ \Gamma(m+\frac{1}{2}) \biggr] \biggl[ \Gamma(m+1) \biggr]^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\sqrt{\pi} }{ (2\chi)^{m+\frac{1}{2}} } \cdot \biggl[\frac{\sqrt{\pi}}{2^m} \cdot (2m-1)!! \biggr] \biggl[ m (m-1)! \biggr]^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{\pi }{2^{2m+\frac{1}{2}} \chi^{m+\frac{1}{2}} } \cdot \biggl[ \frac{(2m-1)!! }{ m (m-1)! }\biggr]
			</math>
			</td>
			</tr>
			</table>


			<table border="1" align="center" cellpadding="8">
			<tr><th align="center" colspan="2">
			Asymptotic behavior:
			</th></tr>
			<tr>
			<td align="center"><math>~m</math></td>
			<td align="center" colspan="1">
			<math>~\lim_{\chi\rightarrow \infty} Q_{m-\frac{1}{2}}(\chi)</math>
			</td>
			</tr>
			<tr>
			<td align="center">0</td>
			<td align="center">
			<math>~
			\frac{\Gamma(\frac{1}{2}) \sqrt{\pi} }{ \Gamma(1) (2\chi)^{\frac{1}{2}} }
			= \frac{\pi}{ (2\chi)^{\frac{1}{2}} } \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center">1</td>
			<td align="center">
			<math>~
			\frac{\pi }{2^{\frac{5}{2}} \chi^{\frac{3}{2}} }
			</math>
			</td>
			</tr>
			<tr>
			<td align="center">2</td>
			<td align="center">
			<math>~
			\frac{\pi }{2^{\frac{9}{2}} \chi^{\frac{5}{2}} } \cdot \biggl[ \frac{(3)!! }{ 2 }\biggr]
			</math>
			</td>
			</tr>
			</table>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T17:49:45Z

Wong Toroidal Coordinates

← Older revision		Revision as of 17:49, 28 June 2018
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	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.

			ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [https://books.google.com/books?id=MtU8uP7XMvoC&printsec=frontcover&dq=Abramowitz+and+stegun&hl=en&sa=X&ved=0ahUKEwialra5xNbaAhWKna0KHcLAASAQ6AEILDAA#v=onepage&q=Abramowitz%20and%20stegun&f=false Abramowitz & Stegun (1995)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>, which is entirely consistent with the lower-precision value that we have extracted from [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].
	</td>		</td>
	</tr>		</tr>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T17:31:19Z

Wong Toroidal Coordinates

← Older revision		Revision as of 17:31, 28 June 2018
Line 328:		Line 328:
	'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.		'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.

	For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in CRC.		For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of the 1971 (19<sup>th</sup>) edition of the CRC's ''Standard Mathematical Tables'', published by the Chemical Rubber Co., Cleveland, Ohio, U.S.A..

	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T17:27:37Z

Wong Toroidal Coordinates

← Older revision		Revision as of 17:27, 28 June 2018
Line 329:		Line 329:

	For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in CRC.		For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in CRC.

			NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T17:13:45Z

Wong Toroidal Coordinates

← Older revision		Revision as of 17:13, 28 June 2018
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	<table align="center" cellpadding="5" border="1">		<table align="center" cellpadding="5" border="1">
	<tr>		<tr>
	<td align="center"><math>~P^0_{-\frac{1}{2}}</math></td>		<td align="center"><math>~P^0_{-\frac{1}{2}}(z)</math></td>
	<td align="center"><math>~P^0_{+\frac{1}{2}}</math></td>		<td align="center"><math>~P^0_{+\frac{1}{2}}(z)</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<table align="center" cellpadding="5" border="1">		<table align="center" cellpadding="5" border="1">
	<tr>		<tr>
	<td align="center"><math>~Q^0_{-\frac{1}{2}}</math></td>		<td align="center"><math>~Q^0_{-\frac{1}{2}}(z)</math></td>
	<td align="center"><math>~Q^0_{+\frac{1}{2}}</math></td>		<td align="center"><math>~Q^0_{+\frac{1}{2}}(z)</math></td>
	<td align="center"><math>~Q^0_{+\frac{3}{2}}</math></td>		<td align="center"><math>~Q^0_{+\frac{3}{2}}(z)</math></td>
	</tr>		</tr>
	<tr>		<tr>
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	<td align="left">		<td align="left">
	'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.		'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.

			For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in CRC.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T16:55:18Z

Wong Toroidal Coordinates

← Older revision		Revision as of 16:55, 28 June 2018
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	<td align="center">		<td align="center">
	[[File:P0plusHalf.png\|250px\|P0plusHalf]]		[[File:P0plusHalf.png\|250px\|P0plusHalf]]
			</td>
			</tr>
			<tr>
			<td align="left" colspan="2">
			See [[#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>
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	<td align="center">		<td align="center">
	[[File:Q0plus3Half.png\|250px\|P0plusHalf]]		[[File:Q0plus3Half.png\|250px\|P0plusHalf]]
			</td>
			</tr>
			<tr>
			<td align="left" colspan="3">
			See [[#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>

Tohline: /* Wong Toroidal Coordinates */

2018-06-28T16:49:40Z

Wong Toroidal Coordinates

← Older revision		Revision as of 16:49, 28 June 2018
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	</tr>		</tr>
	</table>		</table>

			<div align="center" id="Caption">
			<table border="1" cellpadding="8" width="85%" align="center">
			<tr>
			<td align="left">
			'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.
			</td>
			</tr>
			</table>
			</div>

← Older revision		Revision as of 17:31, 28 June 2018
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	'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.		'''Caption for Plots:'''   Here we explain how we assembled the various plots that depict the behavior of various associated Legendre (toroidal) functions, specifically, (above, for varying half-integer degrees) <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math>, <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^0_{+\frac{3}{2}}</math>, and (in association with an [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]], for varying order) <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>.

	For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in CRC.		For each choice of the integer indexes, <math>~n \ge 0</math> and <math>~m \ge 0</math>, the relevant plot shows how the function, <math>~X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>~z</math>. The solid green circular markers in each plot identify data that has been pulled directly from tabulated values published in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]. In each plot, the solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>~K(k)</math> and <math>~E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of the 1971 (19<sup>th</sup>) edition of the CRC's ''Standard Mathematical Tables'', published by the Chemical Rubber Co., Cleveland, Ohio, U.S.A..

	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function, in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.