Appendix/Ramblings/Bordeaux - Revision history

Jet53man: /* The Huré, et al (2020) Presentation */

2022-04-15T23:05:54Z

The Huré, et al (2020) Presentation

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	===The Huré, ''et al'' (2020) Presentation===		===The Huré, ''et al'' (2020) Presentation===

	{{~~LSU_WorkInProgress~~}}		{{ SGFworkInProgress }}

	====Notation====		====Notation====

Jet53man at 23:04, 15 April 2022

2022-04-15T23:04:37Z

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	<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>z \rightarrow \coth\eta</math>, in the "[[Appendix/~~EquationTemplates~~#Analytic_Expressions_.26_Plots\|Key Equation]],"		<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>z \rightarrow \coth\eta</math>, in the "[[Appendix/SpecialFunctions#Analytic_Expressions_.26_Plots\|Key Equation]],"

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	Drawing from our [[Appendix/~~EquationTemplates~~#Analytic_Expressions_.26_Plots\|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,		Drawing from our [[Appendix/SpecialFunctions#Analytic_Expressions_.26_Plots\|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,
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	<math>~1.104816977</math>, which matches [[Appendix/~~EquationTemplates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]]		<math>~1.104816977</math>, which matches [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53\|MF53 value]]
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	Now, from our [[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|accompanying table of "Toroidal Function Evaluations"]], it appears as though,		Now, from our [[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|accompanying table of "Toroidal Function Evaluations"]], it appears as though,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<tr><td align="left">		<tr><td align="left">
	From the [[#FirstEvaluations\|above function tabulations & evaluations]] — for example, <math>~ K(k_0) = 1.854074677</math> and <math>~ E(k_0) = 1.350643881</math> — and a [[Appendix/~~EquationTemplates~~#Example_Recurrence_Relations\|separate listing of ''Example Recurrence Relations'']], we have,		From the [[#FirstEvaluations\|above function tabulations & evaluations]] — for example, <math>~ K(k_0) = 1.854074677</math> and <math>~ E(k_0) = 1.350643881</math> — and a [[Appendix/SpecialFunctions#Example_Recurrence_Relations\|separate listing of ''Example Recurrence Relations'']], we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	[[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
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	[[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
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	<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/~~EquationTemplates~~#Example_Recurrence_Relations\|recurrence]] with m = 2:		<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/SpecialFunctions#Example_Recurrence_Relations\|recurrence]] with m = 2:
	<math>~Q_{+\tfrac{3}{2}}(z_0)</math>		<math>~Q_{+\tfrac{3}{2}}(z_0)</math>
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	[[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
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	<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/~~EquationTemplates~~#Example_Recurrence_Relations\|recurrence]] with m = 3, gives:		<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/SpecialFunctions#Example_Recurrence_Relations\|recurrence]] with m = 3, gives:
	<math>~Q_{+\tfrac{5}{2}}(z_0)</math>		<math>~Q_{+\tfrac{5}{2}}(z_0)</math>
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	Now, from our tabulation of [[Appendix/~~EquationTemplates~~#Example_Recurrence_Relations\|example recurrence relations]], we see that,		Now, from our tabulation of [[Appendix/SpecialFunctions#Example_Recurrence_Relations\|example recurrence relations]], we see that,
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Jet53man: /* More General Evaluation of C0 */

2022-04-15T22:58:47Z

More General Evaluation of C0

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	[[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
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	Hence [[Appendix/~~EquationTemplates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>		Hence [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>
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	[[~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
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	Hence [[Appendix/~~EquationTemplates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>		Hence [[Appendix/SpecialFunctions#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>
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	[[Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/SpecialFunctions#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
	</td>		</td>
	<td align="center">		<td align="center">

Jet53man: /* Wong's Expression */

2022-04-15T22:53:06Z

Wong's Expression

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	[[Image:LSUkey.png\|25px\|link=Appendix/~~EquationTemplates~~#Toroidal_Function_Evaluations]]		[[Image:LSUkey.png\|25px\|link=Appendix/SpecialFunctions#Toroidal_Function_Evaluations]]
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Jet53man: /* Wong's Expression */

2022-04-15T22:48:11Z

Wong's Expression

← Older revision		Revision as of 18:48, 15 April 2022
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	<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>~z \rightarrow \coth\eta</math>, in the "[[Appendix/EquationTemplates#Analytic_Expressions_.26_Plots\|Key Equation]],"		<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>z \rightarrow \coth\eta</math>, in the "[[Appendix/EquationTemplates#Analytic_Expressions_.26_Plots\|Key Equation]],"

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	[[Image:~~LSU_Key~~.png\|25px\|link=~~http://www.vistrails.org/index.php/User:Tohline/~~Appendix/EquationTemplates#Toroidal_Function_Evaluations]]		[[Image:LSUkey.png\|25px\|link=Appendix/EquationTemplates#Toroidal_Function_Evaluations]]
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Jet53man at 22:42, 15 April 2022

2022-04-15T22:42:16Z

← Older revision		Revision as of 18:42, 15 April 2022
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	<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>~z \rightarrow \coth\eta</math>, in the "[[Appendix/~~Equation_templates~~#Analytic_Expressions_.26_Plots\|Key Equation]],"		<span id="KeyEquation">And if we make the function-argument substitution,</span> <math>~z \rightarrow \coth\eta</math>, in the "[[Appendix/EquationTemplates#Analytic_Expressions_.26_Plots\|Key Equation]],"

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	[[Image:LSU_Key.png\|25px\|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations]]		[[Image:LSU_Key.png\|25px\|link=http://www.vistrails.org/index.php/User:Tohline/Appendix/EquationTemplates#Toroidal_Function_Evaluations]]
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	Drawing from our [[Appendix/~~Equation_templates~~#Analytic_Expressions_.26_Plots\|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,		Drawing from our [[Appendix/EquationTemplates#Analytic_Expressions_.26_Plots\|accompanying tabulation of ''Toroidal Function Evaluations'']], we have more generally,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	[[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
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	Hence [[Appendix/~~Equation_templates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>		Hence [[Appendix/EquationTemplates#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{-\tfrac{1}{2}}(3)</math>
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	[[~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>		[[EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
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	Hence [[Appendix/~~Equation_templates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>		Hence [[Appendix/EquationTemplates#Comparison_with_Table_IX_from_MF53\|MF53 value]], <math>~Q_{+\tfrac{1}{2}}(3)</math>
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	[[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
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	<math>~1.104816977</math>, which matches [[Appendix/~~Equation_templates~~#Comparison_with_Table_IX_from_MF53\|MF53 value]]		<math>~1.104816977</math>, which matches [[Appendix/EquationTemplates#Comparison_with_Table_IX_from_MF53\|MF53 value]]
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	</table>		</table>
	Now, from our [[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|accompanying table of "Toroidal Function Evaluations"]], it appears as though,		Now, from our [[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|accompanying table of "Toroidal Function Evaluations"]], it appears as though,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<table border="1" align="center" width="80%" cellpadding="10">		<table border="1" align="center" width="80%" cellpadding="10">
	<tr><td align="left">		<tr><td align="left">
	From the [[#FirstEvaluations\|above function tabulations & evaluations]] — for example, <math>~ K(k_0) = 1.854074677</math> and <math>~ E(k_0) = 1.350643881</math> — and a [[Appendix/~~Equation_templates~~#Example_Recurrence_Relations\|separate listing of ''Example Recurrence Relations'']], we have,		From the [[#FirstEvaluations\|above function tabulations & evaluations]] — for example, <math>~ K(k_0) = 1.854074677</math> and <math>~ E(k_0) = 1.350643881</math> — and a [[Appendix/EquationTemplates#Example_Recurrence_Relations\|separate listing of ''Example Recurrence Relations'']], we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	[[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{-\tfrac{1}{2}}(z_0)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	[[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>		[[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q_{+\tfrac{1}{2}}(z_0)</math>
	</td>		</td>
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	<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/~~Equation_templates~~#Example_Recurrence_Relations\|recurrence]] with m = 2:		<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/EquationTemplates#Example_Recurrence_Relations\|recurrence]] with m = 2:
	<math>~Q_{+\tfrac{3}{2}}(z_0)</math>		<math>~Q_{+\tfrac{3}{2}}(z_0)</math>
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	<td align="right">		<td align="right">
	[[Appendix/~~Equation_templates~~#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>		[[Appendix/EquationTemplates#Toroidal_Function_Evaluations\|Appendix Expression:]] <math>~Q^2_{-\tfrac{1}{2}}(z_0)</math>
	</td>		</td>
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	<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/~~Equation_templates~~#Example_Recurrence_Relations\|recurrence]] with m = 3, gives:		<math>~Q^0_{m-\tfrac{1}{2}}</math> [[Appendix/EquationTemplates#Example_Recurrence_Relations\|recurrence]] with m = 3, gives:
	<math>~Q_{+\tfrac{5}{2}}(z_0)</math>		<math>~Q_{+\tfrac{5}{2}}(z_0)</math>
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	Now, from our tabulation of [[Appendix/~~Equation_templates~~#Example_Recurrence_Relations\|example recurrence relations]], we see that,		Now, from our tabulation of [[Appendix/EquationTemplates#Example_Recurrence_Relations\|example recurrence relations]], we see that,
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Jet53man at 22:33, 15 April 2022

2022-04-15T22:33:05Z

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Jet53man: Created page with "FORCETOC =Université de Bordeaux (Part 1)= ==Spheroid-Ring Systems==..."

2021-07-24T22:48:50Z

Created page with "__FORCETOC__   =Université de Bordeaux (Part 1)= ==Spheroid-Ring Systems==..."

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Appendix/Ramblings/Bordeaux - Revision history

Jet53man: /* The Huré, et al (2020) Presentation */

Jet53man at 23:04, 15 April 2022

Jet53man: /* More General Evaluation of C0 */

Jet53man: /* Wong's Expression */

Jet53man: /* Wong's Expression */

Jet53man at 22:42, 15 April 2022

Jet53man at 22:33, 15 April 2022

Jet53man: Created page with "__FORCETOC__ =Université de Bordeaux (Part 1)= ==Spheroid-Ring Systems==..."

Jet53man: Created page with "FORCETOC =Université de Bordeaux (Part 1)= ==Spheroid-Ring Systems==..."