User:Tohline/Appendix/Mathematics/ToroidalSynopsis01 - Revision history

Tohline: /* Evaluating Q2ν */

2018-09-26T03:48:20Z

Evaluating Q2ν

← Older revision		Revision as of 03:48, 26 September 2018
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			Let's push forward a bit more; specifically, let's find the expressions that are relevant when <math>~n = +1</math>. When <math>~\mu = 0</math>,

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

			<tr>
			<td align="right">
			<math>~Q_{+ \frac{1}{2}}^{1}(z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2} (z^2-1)^{-\frac{1}{2}} \biggl\{ z Q_{+\frac{1}{2}}(z) - Q_{ - \frac{1}{2}}(z) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2} (z^2-1)^{-\frac{1}{2}} \biggl\{ z^2 k~K ( k ) ~-~ [2z^2(z+1)]^{1 / 2} E( k ) ~-~ k K(k)\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2} \biggl\{ (z^2 - 1)^{1 / 2} k~K ( k ) ~-~ \biggl[ \frac{2z^2(z+1)}{z^2-1} \biggr]^{1 / 2} E( k ) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2} \biggl[ (z^2 - 1)^{1 / 2} k~K ( k ) ~-~ \biggl( \frac{2z^2}{z-1} \biggr)^{1 / 2} E( k ) \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

			And, when <math>~\mu = 1</math>,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~Q_{+ \frac{1}{2}}^{2}(z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\frac{1}{2}(z^2-1)^{-\frac{1}{2}} \biggl\{ z Q^1_{+\frac{1}{2}}(z) ~+~3Q^1_{- \frac{1}{2}}(z) \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\frac{1}{2}(z^2-1)^{-\frac{1}{2}} \biggl\{ \frac{z}{2} \biggl[ (z^2 - 1)^{1 / 2} k~K ( k ) ~-~ \biggl( \frac{2z^2}{z-1} \biggr)^{1 / 2} E( k ) \biggr]
			~-~3 \biggl[\frac{1}{2(z-1)} \biggr]^{1 / 2} E(k)\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\frac{1}{2^2}(z^2-1)^{-\frac{1}{2}} \biggl\{ z (z^2 - 1)^{1 / 2} k~K ( k )
			~-~(z^2+3) \biggl( \frac{2}{z-1} \biggr)^{1 / 2} E(k)\biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-~\frac{1}{2^2}\biggl\{ z k~K ( k )
			~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k)\biggr\} \, .
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Evaluating Q2ν */

2018-07-01T17:42:49Z

Evaluating Q2ν

← Older revision		Revision as of 17:42, 1 July 2018
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	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	See [[User:Tohline/Appendix/~~Mathematics/ToroidalFunctions~~#Caption\|relevant caption]].		See [[User:Tohline/Appendix/Equation_templates#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>

Tohline: /* Evaluating Q2ν */

2018-06-30T22:42:00Z

Evaluating Q2ν

← Older revision		Revision as of 22:42, 30 June 2018
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:~~Q1minusHalf~~.png\|250px\|P0minusHalf]]		[[File:ABSQ1minusHalf.png\|250px\|P0minusHalf]]
	</td>		</td>
	<td align="center">		<td align="center">

Tohline: /* Evaluating Q2ν */

2018-06-30T22:10:39Z

Evaluating Q2ν

← Older revision		Revision as of 22:10, 30 June 2018
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	How do we evaluate an "order 2" associated Legendre function, such as, <math>~Q_\nu^2</math> ?		How do we evaluate an "order 2" associated Legendre function, such as, <math>~Q_\nu^2</math> ?

	Start by recognizing that,		Start by recognizing that, from our identified set of "Key Equations,"

	~~<table border="0" cellpadding="5" align="center">~~		{{ User:Tohline/Math/EQ_Toroidal07 }}

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~P_\nu^~~{~~\mu + 1}(z)</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=<~~/~~math>~~
	</~~td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~(z^2-1)^{-\frac{1~~}{2}~~} \{ (\nu - \mu) z P^\mu_\nu(z) - (\nu + \mu)P^\mu_{\nu - 1}(z)\}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" colspan="3">~~
	[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)], p. 333, eq. (8.5.1)
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~NOTE: <math>~Q_\nu^\mu</math>, as well as <math>~P_\nu^\mu</math>, satisfies this same recurrence relation.~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

	Hence, after adopting the association, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, we have, when <math>~\mu = 0</math>,		Hence, after adopting the association, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, we have, when <math>~\mu = 0</math>,

Tohline: /* Integral Over Radial Coordinate */

2018-06-29T22:53:23Z

Integral Over Radial Coordinate

← Older revision		Revision as of 22:53, 29 June 2018
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	Let's see if we can replicate this integration result. Let's start with the ~~first~~ numbered equation under §3.12 (p. 169) of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)], which is prefaced by the statement, "If <math>~w_\nu^\mu(z)</math> and <math>w_\sigma^\rho(z)</math> denote any solutions of Legendre's differential equation … with the parameters <math>~\nu, \mu</math> and <math>~\sigma, \rho</math>, respectively, then it follows … that,"		Let's see if we can replicate this integration result. Let's start with the "Key Equation",

			{{ User:Tohline/Math/EQ_Toroidal05 }}

			<!--numbered equation under §3.12 (p. 169) of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi (1953)], which is prefaced by the statement, "If <math>~w_\nu^\mu(z)</math> and <math>w_\sigma^\rho(z)</math> denote any solutions of Legendre's differential equation … with the parameters <math>~\nu, \mu</math> and <math>~\sigma, \rho</math>, respectively, then it follows … that,"
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			-->

	In order to match the left-hand side of Wong's expression, we should adopt the associations:   <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>~\rho \rightarrow 0</math>, and <math>~\sigma \rightarrow ( n - \tfrac{1}{2})</math>. In which case, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,		In order to match the left-hand side of Wong's expression, we should adopt the associations:   <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 2</math>, <math>~\nu \rightarrow (n - \tfrac{1}{2})</math>, <math>~\rho \rightarrow 0</math>, and <math>~\sigma \rightarrow ( n - \tfrac{1}{2})</math>. In which case, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,

Tohline: /* Evaluating Q2ν */

2018-06-28T16:52:58Z

Evaluating Q2ν

← Older revision		Revision as of 16:52, 28 June 2018
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	<td align="center">		<td align="center">
	[[File:Q2minusHalf.png\|250px\|P0plusHalf]]		[[File:Q2minusHalf.png\|250px\|P0plusHalf]]
			</td>
			</tr>
			<tr>
			<td align="left" colspan="2">
			See [[User:Tohline/Appendix/Mathematics/ToroidalFunctions#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>

Tohline: /* Evaluating Q2ν */

2018-06-28T16:39:28Z

Evaluating Q2ν

← Older revision		Revision as of 16:39, 28 June 2018
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	This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].		This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].

	<div align="center" id="~~Summary~~">		<div align="center" id="Q1Q2Summary">
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>

Tohline: /* Evaluating Q2ν */

2018-06-28T16:31:39Z

Evaluating Q2ν

← Older revision		Revision as of 16:31, 28 June 2018
Line 1,973:		Line 1,973:
	This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].		This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].

			<div align="center" id="Summary">
	<table border="1" align="center" cellpadding="8">		<table border="1" align="center" cellpadding="8">
	<tr>		<tr>
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">


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

	=See Also=		=See Also=

Tohline: /* Evaluating Q2ν */

2018-06-28T16:20:23Z

Evaluating Q2ν

← Older revision		Revision as of 16:20, 28 June 2018
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	<td align="center" colspan="3">		<td align="center" colspan="3">
	where:   <math>~k = \sqrt{ \frac{2}{z+1}} \, .</math>		where:   <math>~k = \sqrt{ \frac{2}{z+1}} \, .</math>
			</td>
			</tr>
			</table>

			<table align="center" cellpadding="5" border="1">
			<tr>
			<td align="center"><math>~\| ~Q^1_{-\frac{1}{2}}(z) ~\|</math></td>
			<td align="center"><math>~Q^2_{-\frac{1}{2}}(z)</math></td>
			</tr>
			<tr>
			<td align="center">
			[[File:Q1minusHalf.png\|250px\|P0minusHalf]]
			</td>
			<td align="center">
			[[File:Q2minusHalf.png\|250px\|P0plusHalf]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Evaluating Q2ν */

2018-06-28T03:30:27Z

Evaluating Q2ν

← Older revision		Revision as of 03:30, 28 June 2018
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	\frac{ 4 z E(k) - (z-1) K(k) }{ [2^{3} (z+1) (z-1)^{2} ]^{1 / 2}} \, .		\frac{ 4 z E(k) - (z-1) K(k) }{ [2^{3} (z+1) (z-1)^{2} ]^{1 / 2}} \, .
	</math>		</math>
			</td>
			</tr>
			</table>
			This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].

			<table border="1" align="center" cellpadding="8">
			<tr>
			<th align="center">Summary</th>
			</tr>
			<tr>
			<td align="left">

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


			<tr>
			<td align="right">
			<math>~Q^1_{-\frac{1}{2}}(z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\frac{1}{2(z-1)} \biggr]^{1 / 2} E(k)
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~Q^2_{-\frac{1}{2}}(z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{ 4 z E(k) - (z-1) K(k) }{ [2^{3} (z+1) (z-1)^{2} ]^{1 / 2}}
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			where:   <math>~k = \sqrt{ \frac{2}{z+1}} \, .</math>
			</td>
			</tr>
			</table>

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