Appendix/Mathematics/ToroidalFunctions - Revision history

Jet53man: /* Summary of Toroidal Coordinates and Toroidal Functions */

2022-04-24T21:14:22Z

Summary of Toroidal Coordinates and Toroidal Functions

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	[[File:Q0minus1Half.png\|250px\|~~P0minusHalf~~]]		[[File:Q0minus1Half.png\|250px\|Q0minus1Half]]
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	[[File:Q0plus1Half.png\|250px\|~~P0plusHalf~~]]		[[File:Q0plus1Half.png\|250px\|Q0plus1Half]]
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	[[File:Q0plus3Half.png\|250px\|~~P0plusHalf~~]]		[[File:Q0plus3Half.png\|250px\|Q0plus3Half]]
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Jet53man: /* Third Dabbling */

2021-07-26T16:49:52Z

Third Dabbling

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	And from our [[~~User:Tohline/~~Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations\|examination of the summation expression]], we have found that,		And from our [[Appendix/Mathematics/ToroidalConfusion#Joel.27s_Additional_Manipulations\|examination of the summation expression]], we have found that,
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	How does this jive with [[~~User:Tohline/~~Appendix/Mathematics/ToroidalSynopsis01#SummaryTable\|our separate recognition]] that,		How does this jive with [[Appendix/Mathematics/ToroidalSynopsis01#SummaryTable\|our separate recognition]] that,
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Jet53man: /* Second Dabbling */

2021-07-26T16:48:57Z

Second Dabbling

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	This expression matches the expression for <math>~\Chi</math>, as defined in the CT99; see, for example, [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|our accompanying discussion]],		This expression matches the expression for <math>~\Chi</math>, as defined in the CT99; see, for example, [[2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|our accompanying discussion]],
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	In a [[~~User:Tohline/~~Appendix/Mathematics/ToroidalSynopsis01#SummaryTable\|separate discussion]], we have derived expressions for the quantities, <math>~\Chi</math> and <math>~\mu^2</math>, in terms of Wong's toroidal coordinates.		In a [[Appendix/Mathematics/ToroidalSynopsis01#SummaryTable\|separate discussion]], we have derived expressions for the quantities, <math>~\Chi</math> and <math>~\mu^2</math>, in terms of Wong's toroidal coordinates.
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Jet53man: /* First Dabbling */

2021-07-26T16:47:48Z

First Dabbling

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	and, <math>~Q_{m - 1 / 2}</math> is the zero-order, half-(odd)integer degree, Llegendre function of the second kind — also referred to as a ''toroidal'' function of zeroth order.		and, <math>~Q_{m - 1 / 2}</math> is the zero-order, half-(odd)integer degree, Llegendre function of the second kind — also referred to as a ''toroidal'' function of zeroth order.

	From the [[~~User:Tohline/~~Apps/DysonWongTori#Introducing_Toroidal_Coordinates\|Dyson-Wong Toroid]] chapter,		From the [[Apps/DysonWongTori#Introducing_Toroidal_Coordinates\|Dyson-Wong Toroid]] chapter,
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Jet53man: /* Drawn From Discussion of Solving the Poisson Equation */

2021-07-26T16:46:47Z

Drawn From Discussion of Solving the Poisson Equation

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	=Drawn From Discussion of Solving the Poisson Equation=		=Drawn From Discussion of Solving the Poisson Equation=
	The following has been copied (May 2018) from an [[~~User:Tohline/~~AxisymmetricConfigurations/PoissonEq#Toroidal_Functions\|accompanying chapter]] that presents the ''integral representation'' of the Poisson equation in terms of toroidal functions.		The following has been copied (May 2018) from an [[AxisymmetricConfigurations/PoissonEq#Toroidal_Functions\|accompanying chapter]] that presents the ''integral representation'' of the Poisson equation in terms of toroidal functions.

	<!-- NEW TABLE 3 -->		<!-- NEW TABLE 3 -->

Jet53man: /* Overview by Howard Cohl */

2021-07-26T16:45:55Z

Overview by Howard Cohl

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	Note that, as shown above, [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G Gil, ''et al.'' (2000)] state:		Note that, as shown above, [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G Gil, ''et al.'' (2000)] state:

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

	Hence, if we swap the indexes, <math>~m\leftrightarrow n</math>, and make the assignments, <math>~x \leftrightarrow \cosh\alpha</math> and <math>~\lambda \leftrightarrow \coth\alpha</math>, this last relation becomes,		Hence, if we swap the indexes, <math>~m\leftrightarrow n</math>, and make the assignments, <math>~x \leftrightarrow \cosh\alpha</math> and <math>~\lambda \leftrightarrow \coth\alpha</math>, this last relation becomes,
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	However, in our situation the so-called "Euler reflection formula for gamma functions" gives the relation,		However, in our situation the so-called "Euler reflection formula for gamma functions" gives the relation,

	{{ ~~User:Tohline/~~Math/EQ_Gamma01 }}		{{ Math/EQ_Gamma01 }}

	that is,		that is,

Jet53man: /* Toroidal Functions */

2021-07-26T16:44:38Z

Toroidal Functions

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	Relationship between one another, as per equation (8) in [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204 - 217)]:		Relationship between one another, as per equation (8) in [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G A. Gil, J. Segura, & N. M. Temme (2000, JCP, 161, 204 - 217)]:

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

	Note that the relationship between <math>~\lambda</math> and <math>~x</math> is the same as the relationship between <math>~\cosh\alpha</math> and <math>~\coth\alpha</math>, that is,		Note that the relationship between <math>~\lambda</math> and <math>~x</math> is the same as the relationship between <math>~\cosh\alpha</math> and <math>~\coth\alpha</math>, that is,
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	===PminusHalf01===		===PminusHalf01===

	{{ ~~User:Tohline/~~Math/EQ_PminusHalf01 }}		{{ Math/EQ_PminusHalf01 }}

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	===QminusHalf01===		===QminusHalf01===

	{{ ~~User:Tohline/~~Math/EQ_QminusHalf01 }}		{{ Math/EQ_QminusHalf01 }}

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	===PplusHalf01===		===PplusHalf01===

	{{ ~~User:Tohline/~~Math/EQ_PplusHalf01 }}		{{ Math/EQ_PplusHalf01 }}

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	===QplusHalf01===		===QplusHalf01===

	{{ ~~User:Tohline/~~Math/EQ_QplusHalf01 }}		{{ Math/EQ_QplusHalf01 }}

Jet53man: /* CT99 Coordinates */

2021-07-26T16:42:19Z

CT99 Coordinates

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	==CT99 Coordinates==		==CT99 Coordinates==

	<font color="red"><b>Eureka!</b></font> Via his dogged efforts and an extraordinarily in-depth investigation of this problem, [[~~User:Tohline/~~Appendix/Ramblings/CCGF#Compact_Cylindrical_Green_Function_.28CCGF.29\|in 1999 Howard S. Cohl discovered]] that, in cylindrical coordinates, the relevant Green's function can be written in a much more compact and much more practical form. Specifically, he realized that,		<font color="red"><b>Eureka!</b></font> Via his dogged efforts and an extraordinarily in-depth investigation of this problem, [[Appendix/Ramblings/CCGF#Compact_Cylindrical_Green_Function_.28CCGF.29\|in 1999 Howard S. Cohl discovered]] that, in cylindrical coordinates, the relevant Green's function can be written in a much more compact and much more practical form. Specifically, he realized that,
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	where, <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0=1</math> and <math>~\epsilon_m = 2</math> for <math>~m\ge 1</math>. Following this discovery, most of my research group's 3D numerical modeling of self-gravitating fluids has been conducted using ''Toroidal functions'' instead of ''Spherical Harmonics'' to evaluate the boundary potential on our cylindrical-coordinate meshes.		where, <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0=1</math> and <math>~\epsilon_m = 2</math> for <math>~m\ge 1</math>. Following this discovery, most of my research group's 3D numerical modeling of self-gravitating fluids has been conducted using ''Toroidal functions'' instead of ''Spherical Harmonics'' to evaluate the boundary potential on our cylindrical-coordinate meshes.

	In our [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|accompanying statement of this problem]], we have written,		In our [[2DStructure/ToroidalCoordinates#Statement_of_the_Problem\|accompanying statement of this problem]], we have written,
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Jet53man: /* Summary of Toroidal Coordinates and Toroidal Functions */

2021-07-26T16:40:44Z

Summary of Toroidal Coordinates and Toroidal Functions

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	Expressions for the relevant toroidal functions are as follows:		Expressions for the relevant toroidal functions are as follows:

	{{ ~~User:Tohline/~~Math/EQ_PminusHalf01 }}		{{ Math/EQ_PminusHalf01 }}

	{{ ~~User:Tohline/~~Math/EQ_PplusHalf01 }}		{{ Math/EQ_PplusHalf01 }}

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	See [[~~User:Tohline/~~Appendix/~~Equation_templates~~#Caption\|relevant caption]].		See [[Appendix/EquationTemplates#Caption\|relevant caption]].
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	{{ ~~User:Tohline/~~Math/EQ_QminusHalf01 }}		{{ Math/EQ_QminusHalf01 }}

	{{ ~~User:Tohline/~~Math/EQ_QplusHalf01 }}		{{ Math/EQ_QplusHalf01 }}
	where, <math>~K</math> and <math>~E</math> are complete elliptic integrals of the first and second kind, respectively. In equation (23) of [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)], this last expression has been written in the more compact form,		where, <math>~K</math> and <math>~E</math> are complete elliptic integrals of the first and second kind, respectively. In equation (23) of [http://adsabs.harvard.edu/abs/1999ApJ...527...86C Cohl & Tohline (1999)], this last expression has been written in the more compact form,

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	See [[~~User:Tohline/~~Appendix/~~Equation_templates~~#Caption\|relevant caption]].		See [[Appendix/EquationTemplates#Caption\|relevant caption]].
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	We also will employ the so-called ''recurrence relation'',		We also will employ the so-called ''recurrence relation'',

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

	After setting, <math>~\mu = 0</math>, and making the association, <math>~\nu \rightarrow (m - \tfrac{3}{2})</math>, for example, this gives,		After setting, <math>~\mu = 0</math>, and making the association, <math>~\nu \rightarrow (m - \tfrac{3}{2})</math>, for example, this gives,

Jet53man: Created page with " FORCETOC =Relationships Between Toroidal Functions= ==Wong Toroidal Coordinates== ===Wong's Expression for the Potential=== This c..."

2021-07-26T16:38:03Z

Created page with " __FORCETOC__ =Relationships Between Toroidal Functions= ==Wong Toroidal Coordinates== ===Wong's Expression for the Potential=== This c..."

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Appendix/Mathematics/ToroidalFunctions - Revision history

Jet53man: /* Summary of Toroidal Coordinates and Toroidal Functions */

Jet53man: /* Third Dabbling */

Jet53man: /* Second Dabbling */

Jet53man: /* First Dabbling */

Jet53man: /* Drawn From Discussion of Solving the Poisson Equation */

Jet53man: /* Overview by Howard Cohl */

Jet53man: /* Toroidal Functions */

Jet53man: /* CT99 Coordinates */

Jet53man: /* Summary of Toroidal Coordinates and Toroidal Functions */

Jet53man: Created page with " __FORCETOC__ =Relationships Between Toroidal Functions= ==Wong Toroidal Coordinates== ===Wong's Expression for the Potential=== This c..."

Jet53man: Created page with " FORCETOC =Relationships Between Toroidal Functions= ==Wong Toroidal Coordinates== ===Wong's Expression for the Potential=== This c..."