2DStructure/ToroidalGreenFunction - Revision history

Jet53man: /* See Also */

2021-09-13T22:44:44Z

Jet53man: /* See Also */

2021-09-13T21:50:36Z

Jet53man: /* As Presented by Hicks (1881) */

2021-09-11T20:50:10Z

As Presented by Hicks (1881)

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	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">
	<tr>		<tr>
	<td align="center">[[File:Hicks1881TitlePage.png\|500px\|Title Page of Hicks (1881)]]</td>		<td align="center">
			<!-- [[File:Hicks1881TitlePage.png\|500px\|Title Page of Hicks (1881)]] -->
			[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
			''"On Toroidal Functions"''<br />
			Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652
			</td>
	</tr>		</tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:Hicks1881GreenFunction.png\|750px\|~~Green's Function~~ from Hicks (1881)]]		[[File:Hicks1881GreenFunction.png\|750px\|To be inserted: Eq. (35) from Hicks (1881)]]
	</td></tr></table>		</td></tr></table>

Jet53man: /* Basic Elements of a Toroidal Coordinate System */

2021-09-11T20:48:48Z

Basic Elements of a Toroidal Coordinate System

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	[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />		[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
	''"On Toroidal Functions"''<br />		''"On Toroidal Functions"''<br />
	Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652~~<br />~~		Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652

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

Jet53man: /* Basic Elements of a Toroidal Coordinate System */

2021-09-11T20:41:30Z

Basic Elements of a Toroidal Coordinate System

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	<table border="1" cellpadding="10" align="center" width="85%">		<table border="1" cellpadding="10" align="center" width="85%">
	<tr>		<tr>
	<td align="center">[[File:Hicks1881TitlePage.png\|500px\|Title Page of Hicks (1881)]]</td>		<td align="center">
			<!-- [[File:Hicks1881TitlePage.png\|500px\|Title Page of Hicks (1881)]] -->
			[https://ui.adsabs.harvard.edu/abs/1881RSPT..172..609H/abstract W. M. Hicks (1881)]<br />
			''"On Toroidal Functions"''<br />
			Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609-652<br />

			</td>
	</tr>		</tr>
	<tr><td align="left">		<tr><td align="left">

Jet53man: /* Appendix A: Selected Toroidal Function Relationships */

2021-07-26T16:32:28Z

Appendix A: Selected Toroidal Function Relationships

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	=Appendix A: Selected Toroidal Function Relationships=		=Appendix A: Selected Toroidal Function Relationships=

	Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our [[Appendix/~~Equation_templates~~#Torus\|accompanying ''Equations'' appendix]]. Two related appendices — numbered '''A.3''' and '''A.4''' — may be found in an [[Apps/Wong1973Potential#Appendix:_Selected_Toroidal_Function_Relationships\|accompanying discussion]].		Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our [[Appendix/EquationTemplates#Torus\|accompanying ''Equations'' appendix]]. Two related appendices — numbered '''A.3''' and '''A.4''' — may be found in an [[Apps/Wong1973Potential#Appendix:_Selected_Toroidal_Function_Relationships\|accompanying discussion]].

	==A.1==		==A.1==

Jet53man: /* Series Expansions */

2021-07-26T16:30:59Z

Series Expansions

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	==Series Expansions==		==Series Expansions==

	From our [[~~User:Tohline/~~Appendix/~~Equation_templates~~#Complete_Elliptic_Integrals\|accompanying set of Key mathematical relations]] we find the following series expansions for complete elliptic integrals of, respectively, the first and second kind:		From our [[Appendix/EquationTemplates#Complete_Elliptic_Integrals\|accompanying set of Key mathematical relations]] we find the following series expansions for complete elliptic integrals of, respectively, the first and second kind:

	<div align="center">		<div align="center">
	{{ ~~User:Tohline/~~Math/EQ_EllipticIntegral01 }}<br />		{{ Math/EQ_EllipticIntegral01 }}<br />
	{{ ~~User:Tohline/~~Math/EQ_EllipticIntegral02 }}		{{ Math/EQ_EllipticIntegral02 }}
	</div>		</div>
	If you don't know how to handle a ''double factorial'' — also referred to as a ''semifactorial'' — see, for example, [https://en.wikipedia.org/wiki/Double_factorial this wikipedia page].		If you don't know how to handle a ''double factorial'' — also referred to as a ''semifactorial'' — see, for example, [https://en.wikipedia.org/wiki/Double_factorial this wikipedia page].

Jet53man: /* Summary Table */

2021-07-26T16:29:55Z

Summary Table

← Older revision		Revision as of 12:29, 26 July 2021
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	=Appendix B: Elliptic Integrals=		=Appendix B: Elliptic Integrals=
	==Summary Table==		==Summary Table==
	If the default argument of a [[~~User:Tohline/~~Appendix/~~Equation_templates~~#Complete_Elliptic_Integrals\|complete elliptic integral function]] is the parameter, <math>~k</math>, then we should also appreciate that the following additional arguments may be useful:		If the default argument of a [[Appendix/EquationTemplates#Complete_Elliptic_Integrals\|complete elliptic integral function]] is the parameter, <math>~k</math>, then we should also appreciate that the following additional arguments may be useful:

	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">

Jet53man: /* Appendix A: Selected Toroidal Function Relationships */

2021-07-26T16:16:03Z

Appendix A: Selected Toroidal Function Relationships

← Older revision		Revision as of 12:16, 26 July 2021
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	=Appendix A: Selected Toroidal Function Relationships=		=Appendix A: Selected Toroidal Function Relationships=

	Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our [[~~User:Tohline/~~Appendix/Equation_templates#Torus\|accompanying ''Equations'' appendix]]. Two related appendices — numbered '''A.3''' and '''A.4''' — may be found in an [[~~User:Tohline/~~Apps/Wong1973Potential#Appendix:_Selected_Toroidal_Function_Relationships\|accompanying discussion]].		Here, we draw from the set of toroidal function relationships that have been identified as "Key Equations" in our [[Appendix/Equation_templates#Torus\|accompanying ''Equations'' appendix]]. Two related appendices — numbered '''A.3''' and '''A.4''' — may be found in an [[Apps/Wong1973Potential#Appendix:_Selected_Toroidal_Function_Relationships\|accompanying discussion]].

	==A.1==		==A.1==
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	<td align="left">		<td align="left">
	Beginning with the identified "Key Equation",		Beginning with the identified "Key Equation",
	{{ ~~User:Tohline/~~Math/EQ_Toroidal02 }}		{{ Math/EQ_Toroidal02 }}

	we'll identify <math>~x</math> with <math>~\cosh\eta</math> — in which case we have <math>~\lambda = \coth\eta</math> — and switch the index notations, <math>~n \leftrightarrow m</math>. This gives,		we'll identify <math>~x</math> with <math>~\cosh\eta</math> — in which case we have <math>~\lambda = \coth\eta</math> — and switch the index notations, <math>~n \leftrightarrow m</math>. This gives,
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	Drawing upon the ''Euler reflection formula for gamma functions'', namely,		Drawing upon the ''Euler reflection formula for gamma functions'', namely,
	{{ ~~User:Tohline/~~Math/EQ_Gamma01 }}		{{ Math/EQ_Gamma01 }}

	where it is understood that <math>~m</math> and <math>~n</math> are each either zero or a positive integer, this toroidal-function relation becomes,		where it is understood that <math>~m</math> and <math>~n</math> are each either zero or a positive integer, this toroidal-function relation becomes,
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	<td align="left">		<td align="left">
	Again, beginning with the identified "Key Equation",		Again, beginning with the identified "Key Equation",
	{{ ~~User:Tohline/~~Math/EQ_Toroidal02 }}		{{ Math/EQ_Toroidal02 }}

	this time, without switching index notations, we'll identify <math>~x</math> with <math>~\coth\eta</math> — in which case we have <math>~\lambda = \cosh\eta</math>. This gives,		this time, without switching index notations, we'll identify <math>~x</math> with <math>~\coth\eta</math> — in which case we have <math>~\lambda = \cosh\eta</math>. This gives,
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	Finally, calling upon the "Key Equation" relation,		Finally, calling upon the "Key Equation" relation,

	{{ ~~User:Tohline/~~Math/EQ_Toroidal00 }}		{{ Math/EQ_Toroidal00 }}

	making the index notation substitution, <math>~\nu \rightarrow (m-\tfrac{1}{2})</math>, and associating <math>~z</math> with <math>~ \coth\eta</math> gives,		making the index notation substitution, <math>~\nu \rightarrow (m-\tfrac{1}{2})</math>, and associating <math>~z</math> with <math>~ \coth\eta</math> gives,
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	Drawing from Volume I of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi's (1953)] ''Higher Transcendental Functions,'' we find the following "Key Equation":		Drawing from Volume I of [https://authors.library.caltech.edu/43491/1/Volume%201.pdf A. Erdélyi's (1953)] ''Higher Transcendental Functions,'' we find the following "Key Equation":

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

	Next we adopt the associations, <math>~z \rightarrow \cosh\eta</math>, <math>~t \rightarrow \theta</math>, <math>~\mu \rightarrow 2</math>, and, <math>~\nu \rightarrow n - \tfrac{1}{2}</math>, where <math>~n</math> is zero or a positive integer. In this case we have,		Next we adopt the associations, <math>~z \rightarrow \cosh\eta</math>, <math>~t \rightarrow \theta</math>, <math>~\mu \rightarrow 2</math>, and, <math>~\nu \rightarrow n - \tfrac{1}{2}</math>, where <math>~n</math> is zero or a positive integer. In this case we have,
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	Beginning with the identified "Key Equation",		Beginning with the identified "Key Equation",

	{{ ~~User:Tohline/~~Math/EQ_Toroidal05 }}		{{ Math/EQ_Toroidal05 }}

	we will 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>. As a result, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,		we will 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>. As a result, [https://authors.library.caltech.edu/43491/1/Volume%201.pdf Erdélyi's (1953)] expression becomes,
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	Drawing upon the ''recurrence'' "Key Equation,"		Drawing upon the ''recurrence'' "Key Equation,"

	{{ ~~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,

Jet53man: /* Uniform-Density Torus */

2021-07-26T16:09:47Z

Uniform-Density Torus

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	==Uniform-Density Torus==		==Uniform-Density Torus==

	In an [[~~User:Tohline/~~Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|accompanying discussion]], we build upon the above technical foundation and detail how [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, has heretofore been underappreciated.		In an [[Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential\|accompanying discussion]], we build upon the above technical foundation and detail how [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] was able to complete both integrals to derive an analytic expression for the potential (inside as well as outside) of axisymmetric, uniform-density tori having an arbitrarily specified ratio of the major to minor (cross-sectional) radii, <math>~R/d</math>. This is an outstanding accomplishment that has received little attention in the astrophysics literature and, therefore, has heretofore been underappreciated.

	=Appendix A: Selected Toroidal Function Relationships=		=Appendix A: Selected Toroidal Function Relationships=

← Older revision		Revision as of 18:44, 13 September 2021
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	<li>		<li>
	(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]		(Apollonian Circles) [[2DStructure/ToroidalCoordinateIntegrationLimits\|Toroidal-Coordinates Integration Limits]]
			</li>
			<li>
			[[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems\|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
	</li>		</li>
	</ul>		</ul>

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 If you don't know how to handle a ''double factorial'' &#8212; also referred to as a ''semifactorial'' &#8212; see, for example, [https://en.wikipedia.org/wiki/Double_factorial this wikipedia page].
-=See Also=
+=By The Way &hellip;=
 <ul>
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 </li>
 </ul>
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