Appendix/Mathematics/ToroidalSynopsis01 - Revision history

Jet53man: /* Evaluating Q2ν */

2022-04-24T21:42:22Z

Evaluating Q2ν

← Older revision		Revision as of 17:42, 24 April 2022
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:ABSQ1minusHalf.png\|250px\|~~P0minusHalf~~]]		[[File:ABSQ1minusHalf.png\|250px\|ABSQ1minusHalf]]
	</td>		</td>
	<td align="center">		<td align="center">
	[[File:Q2minusHalf.png\|250px\|~~P0plusHalf~~]]		[[File:Q2minusHalf.png\|250px\|Q2minusHalf]]
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	See [[Appendix/~~EquationTemplates~~#~~Caption~~\|relevant caption]].		See [[Appendix/SpecialFunctions##Caption_for_Plots\|relevant caption]].
	</td>		</td>
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Jet53man at 21:39, 24 April 2022

2022-04-24T21:39:42Z

← Older revision		Revision as of 17:39, 24 April 2022
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	Equation (10) in §3.7, p. 156 of Volume I gives,		Equation (10) in §3.7, p. 156 of Volume I gives,

	{{ ~~User:Tohline/~~Math/EQ_Toroidal03 }}		{{ Math/EQ_Toroidal03 }}

	</li>		</li>
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	Let's see if we can replicate this integration result. Let's start with the "Key Equation",		Let's see if we can replicate this integration result. Let's start with the "Key Equation",

	{{ ~~User:Tohline/~~Math/EQ_Toroidal05 }}		{{ 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,"		<!--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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	<font color="maroon">'''Reconciliation Attempt #1:'''</font>   Keeping in mind that,		<font color="maroon">'''Reconciliation Attempt #1:'''</font>   Keeping in mind that,

	{{ ~~User:Tohline/~~Math/EQ_Toroidal04 }}		{{ Math/EQ_Toroidal04 }}

	which means, after making the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, that,		which means, after making the associations, <math>~z \rightarrow t</math>, <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (n-\tfrac{1}{2})</math>, that,
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	Start by recognizing that, from our identified set of "Key Equations,"		Start by recognizing that, from our identified set of "Key Equations,"

	{{ ~~User:Tohline/~~Math/EQ_Toroidal07 }}		{{ Math/EQ_Toroidal07 }}

	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>,
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	Therefore, starting from the "Key Equation",		Therefore, starting from the "Key Equation",
	{{ ~~User:Tohline/~~Math/EQ_QminusHalf01 }}		{{ Math/EQ_QminusHalf01 }}
	we associate <math>~k^2 \leftrightarrow 2/(z+1)</math>, which implies,		we associate <math>~k^2 \leftrightarrow 2/(z+1)</math>, which implies,
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	</table>		</table>

	With the exception of the leading negative sign, this 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>].		With the exception of the leading negative sign, this appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].

	Now, let's evaluate the second derivative …		Now, let's evaluate the second derivative …
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	</table>		</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>].		This ''also'' appears to match the tabulated values published in the bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>].

	<div align="center" id="Q1Q2Summary">		<div align="center" id="Q1Q2Summary">
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	<tr>		<tr>
	<td align="left" colspan="2">		<td align="left" colspan="2">
	See [[~~User:Tohline/~~Appendix/~~Equation_templates~~#Caption\|relevant caption]].		See [[Appendix/EquationTemplates#Caption\|relevant caption]].
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Identical Green's Function Expressions */

2021-07-26T17:06:29Z

Identical Green's Function Expressions

← Older revision		Revision as of 13:06, 26 July 2021
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	[https://authors.library.caltech.edu/43491/ Caltech's electronic version] of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi's (1953)] ''Higher Transcendental Functions''; in particular, §3.11, p. 169 of Volume I gives,		[https://authors.library.caltech.edu/43491/ Caltech's electronic version] of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi's (1953)] ''Higher Transcendental Functions''; in particular, §3.11, p. 169 of Volume I gives,
	{{ ~~User:Tohline/~~Math/EQ_Toroidal01 }}		{{ Math/EQ_Toroidal01 }}

	If we make the association, <math>~t \leftrightarrow \coth\eta</math>, then we also have,		If we make the association, <math>~t \leftrightarrow \coth\eta</math>, then we also have,
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	Also, from [[~~User:Tohline/~~Appendix/Mathematics/ToroidalConfusion#QPrelation\|our derived <math>~Q-P</math> relation]],		Also, from [[Appendix/Mathematics/ToroidalConfusion#QPrelation\|our derived <math>~Q-P</math> relation]],
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	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</div>		</div>

	Next, we pull from the [[~~User:Tohline/~~Appendix/Mathematics/ToroidalConfusion#Gil\|accompanying discussion of the Gil et al. (2000) expression]],		Next, we pull from the [[Appendix/Mathematics/ToroidalConfusion#Gil\|accompanying discussion of the Gil et al. (2000) expression]],

	{{ ~~User:Tohline/~~Math/EQ_Toroidal02 }}		{{ Math/EQ_Toroidal02 }}

	Identifying <math>~x</math> with <math>~\cosh\eta</math>, in which case we have <math>~\lambda = \coth\eta</math>, and, switching index notation, <math>~n \leftrightarrow m</math>, gives,		Identifying <math>~x</math> with <math>~\cosh\eta</math>, in which case we have <math>~\lambda = \coth\eta</math>, and, switching index notation, <math>~n \leftrightarrow m</math>, gives,
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	</div>		</div>

	where, this last step also incorporates the [[~~User:Tohline/~~Appendix/Mathematics/ToroidalConfusion#Proper_Interpretation_of_DLMF_Expression\|"Euler reflection formula for gamma functions"]], namely,		where, this last step also incorporates the [[Appendix/Mathematics/ToroidalConfusion#Proper_Interpretation_of_DLMF_Expression\|"Euler reflection formula for gamma functions"]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Green's Function */

2021-07-26T17:04:02Z

Green's Function

← Older revision		Revision as of 13:04, 26 July 2021
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	</li>		</li>
	<li>Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed. In this context, note that the [[~~User:Tohline/~~Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations\|summation expression]] gives,		<li>Wong's expression contains not only a summation over the index, <math>~m</math>, but also an explicit summation over the index, <math>~n</math>, which multiplies the "polar" angle, <math>~\theta</math>; no such additional summation appears in the CT99 expression, indicating that the summation over <math>~n</math> has implicitly already been completed. In this context, note that the [[Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations\|summation expression]] gives,

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

Jet53man: /* Arguments of Q and K */

2021-07-26T17:02:57Z

Arguments of Q and K

← Older revision		Revision as of 13:02, 26 July 2021
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	[[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Also]],		[[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|Also]],

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Jet53man: Created page with " =Synopsis of Toroidal Coordinate Approach= ==Basics== Here we attem..."

2021-07-26T17:02:00Z

Created page with "  =Synopsis of Toroidal Coordinate Approach= ==Basics== Here we attem..."

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