Appendix/SpecialFunctions - Revision history

Jet53man: /* Relationships Between Various Associated Legendre Functions */

2021-07-26T16:25:56Z

Relationships Between Various Associated Legendre Functions

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 <font color="red">Resulting Equation</font>
    </th>
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Jet53man: /* Caption for Plots */

2021-07-18T21:06:50Z

Caption for Plots

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	NOTE: The tabulated values of the function, <math>Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[~~User:Tohline/~~Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — also see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — also see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>\|Q^1_{-\frac{1}{2}}(z)\|</math>.


	ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [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)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>. As can be seen in the [[#Comparison_with_Table_IX_from_MF53\|table of function values immediately below]], this is entirely consistent with the lower-precision value that we have extracted from [<b>[[~~User:Tohline/~~Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], and exactly matches the double-precision value we have calculated based on the [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithm.		ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [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)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>. As can be seen in the [[#Comparison_with_Table_IX_from_MF53\|table of function values immediately below]], this is entirely consistent with the lower-precision value that we have extracted from [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], and exactly matches the double-precision value we have calculated based on the [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithm.
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Caption for Plots */

2021-07-18T21:04:55Z

Caption for Plots

← Older revision		Revision as of 17:04, 18 July 2021
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	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — also see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[~~User:Tohline/~~Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — also see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>\|Q^1_{-\frac{1}{2}}(z)\|</math>.

Jet53man: /* Relationships Between Various Associated Legendre Functions */

2021-07-18T20:28:30Z

Relationships Between Various Associated Legendre Functions

← Older revision		Revision as of 16:28, 18 July 2021
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	<td colspan=2>		<td colspan=2>
	To insert a given equation into any Wiki document, type ...<br /><center>		To insert a given equation into any Wiki document, type ...<br /><center>
	{{ ~~User:Tohline/~~Math/<i><font color="red">Template_Name</font></i> }}</center>		{{ Math/<i><font color="red">Template_Name</font></i> }}</center>
	</td>		</td>
	<td colspan=1 rowspan="2" align="center">		<td colspan=1 rowspan="2" align="center">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal01\|EQ_Toroidal01]]		[[Template:Math/EQ_Toroidal01\|EQ_Toroidal01]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal01}}		{{ Math/EQ_Toroidal01 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
	* [http://adsabs.harvard.edu/abs/1940QJMat..11..222C T. G. Cowling (1940)]:   p. 223 (note sign discrepancy in argument of <math>~Q_\nu</math>)		* [http://adsabs.harvard.edu/abs/1940QJMat..11..222C T. G. Cowling (1940)]:   p. 223 (note sign discrepancy in argument of <math>Q_\nu</math>)
	* [https://dlmf.nist.gov/14.18.E5 DLMF §14.18.5]		* [https://dlmf.nist.gov/14.18.E5 DLMF §14.18.5]
	</td>		</td>
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal02\|EQ_Toroidal02]]		[[Template:Math/EQ_Toroidal02\|EQ_Toroidal02]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal02}}		{{ Math/EQ_Toroidal02 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal03\|EQ_Toroidal03]]		[[Template:Math/EQ_Toroidal03\|EQ_Toroidal03]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal03}}		{{ Math/EQ_Toroidal03 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal04\|EQ_Toroidal04]]		[[Template:Math/EQ_Toroidal04\|EQ_Toroidal04]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal04}}		{{ Math/EQ_Toroidal04 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal05\|EQ_Toroidal05]]		[[Template:Math/EQ_Toroidal05\|EQ_Toroidal05]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal05}}		{{ Math/EQ_Toroidal05 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal06\|EQ_Toroidal06]]		[[Template:Math/EQ_Toroidal06\|EQ_Toroidal06]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal06}}		{{ Math/EQ_Toroidal06 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal07\|EQ_Toroidal07]]		[[Template:Math/EQ_Toroidal07\|EQ_Toroidal07]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal07}}		{{ Math/EQ_Toroidal07 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">
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	<tr>		<tr>
	<td>		<td>
	[[~~User~~:~~Tohline/~~Math/EQ_Toroidal08\|EQ_Toroidal08]]		[[Template:Math/EQ_Toroidal08\|EQ_Toroidal08]]
	</td>		</td>
	<td align="center">		<td align="center">
	{{~~User:Tohline/~~Math/EQ_Toroidal08}}		{{ Math/EQ_Toroidal08 }}
	</td>		</td>
	<td colspan=1 align="left">		<td colspan=1 align="left">

Jet53man: /* Comparison with Table IX from MF53 */

2021-07-18T20:22:06Z

Comparison with Table IX from MF53

← Older revision		Revision as of 16:22, 18 July 2021
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	====Comparison with Table IX from MF53====		====Comparison with Table IX from MF53====

	To facilitate ''copying & pasting'' for immediate use by other researchers, here we present in a tab-delimited, plain-text format the evaluation of nine separate toroidal functions: (''Top half of table'') <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math> and <math>~P^0_{+\frac{3}{2}}</math>; (''Bottom half of table'') <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^1_{+\frac{1}{2}}</math> and <math>~Q^0_{+\frac{3}{2}}</math>. Each function has been evaluated for approximately 23 different argument values in the range, <math>~1.0 \le z \le 9.0</math>, and, for each function, two columns of function values have been presented: (''Left column'') Low-precision evaluation extracted directly from Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]; (''Right column'') Our double-precision evaluation based on a set of ''Numerical Recipes'' algorithms. One exception: The value listed under the "MF53" column for the evaluation of <math>~Q^0_{-\frac{1}{2}}(z=2.6)</math> is the high-precision value published on p. 340 of [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)]; notice that our high-precision evaluation matches all ten digits of their published value.		To facilitate ''copying & pasting'' for immediate use by other researchers, here we present in a tab-delimited, plain-text format the evaluation of nine separate toroidal functions: (''Top half of table'') <math>~P^0_{-\frac{1}{2}}</math>, <math>~P^0_{+\frac{1}{2}}</math> and <math>~P^0_{+\frac{3}{2}}</math>; (''Bottom half of table'') <math>~Q^0_{-\frac{1}{2}}</math>, <math>~Q^1_{-\frac{1}{2}}</math>, <math>~Q^2_{-\frac{1}{2}}</math>, <math>~Q^0_{+\frac{1}{2}}</math>, <math>~Q^1_{+\frac{1}{2}}</math> and <math>~Q^0_{+\frac{3}{2}}</math>. Each function has been evaluated for approximately 23 different argument values in the range, <math>~1.0 \le z \le 9.0</math>, and, for each function, two columns of function values have been presented: (''Left column'') Low-precision evaluation extracted directly from Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]; (''Right column'') Our double-precision evaluation based on a set of [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithms. One exception: The value listed under the "MF53" column for the evaluation of <math>~Q^0_{-\frac{1}{2}}(z=2.6)</math> is the high-precision value published on p. 340 of [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)]; notice that our high-precision evaluation matches all ten digits of their published value.


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	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="left">
	Bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]		<div align="center">Bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]</div>
			<font color="red">ATTENTION:</font>   Widen your browser window, or "zoom out," in order to obtain a proper view of the space-delimited columns of numbers in this table.
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Example Recurrence Relations */

2021-07-18T20:16:22Z

Example Recurrence Relations

← Older revision		Revision as of 16:16, 18 July 2021
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	====Example Recurrence Relations====		====Example Recurrence Relations====

	The above [[#Analytic_Expressions_.26_Plots\|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>~P^0_{-\frac{1}{2}}(z)</math> and <math>~P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>~Q^0_{-\frac{1}{2}}(z)</math> and <math>~Q^0_{+\frac{1}{2}}(z)</math>. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"		The above [[#Analytic_Expressions_.26_Plots\|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>P^0_{-\frac{1}{2}}(z)</math> and <math>P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>Q^0_{-\frac{1}{2}}(z)</math> and <math>Q^0_{+\frac{1}{2}}(z)</math>. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

	{{ Math/EQ_Toroidal04 }}		{{ Math/EQ_Toroidal04 }}

	Specifically, letting <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,		Specifically, letting <math>\mu \rightarrow 0</math> and <math>\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~P^0_~~{m-\frac{1}{2}}(z)</math>		<math>~P0_{m-\frac{1}{2}}(z)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4 \biggl[ \frac{m-1}{2m-1} \biggr] z P^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]P^0_{m-\frac{5}{2}}(z) \, ;</math>     and,		<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z P^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]P^0_{m-\frac{5}{2}}(z) \, ;</math>     and,
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~Q^0_{m-\frac{1}{2}}(z)</math>		<math>Q^0_{m-\frac{1}{2}}(z)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~4 \biggl[ \frac{m-1}{2m-1} \biggr] z Q^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]Q^0_{m-\frac{5}{2}}(z) \, .</math>		<math>4 \biggl[ \frac{m-1}{2m-1} \biggr] z Q^0_{m-\frac{3}{2}}(z) - \biggl[ \frac{2m-3}{2m-1}\biggr]Q^0_{m-\frac{5}{2}}(z) \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	As examples, these two relations have been used to generate columns of numbers in the [[#Comparison_with_Table_IX_from_MF53\|comparison table shown below]] for, respectively, the toroidal functions, <math>~P^0_{+\frac{3}{2}}(z)</math> and <math>~Q^0_{+\frac{3}{2}}(z)</math>. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) <math>~Q^1_{-\frac{1}{2}}(z)</math> and <math>~Q^2_{-\frac{1}{2}}(z)</math>. But, as we have detailed in an [[~~User:Tohline/~~Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD\|accompanying discussion]], additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,		As examples, these two relations have been used to generate columns of numbers in the [[#Comparison_with_Table_IX_from_MF53\|comparison table shown below]] for, respectively, the toroidal functions, <math>P^0_{+\frac{3}{2}}(z)</math> and <math>Q^0_{+\frac{3}{2}}(z)</math>. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) <math>Q^1_{-\frac{1}{2}}(z)</math> and <math>Q^2_{-\frac{1}{2}}(z)</math>. But, as we have detailed in an [[Appendix/Mathematics/ToroidalSynopsis01#Evaluating_Q2.CE.BD\|accompanying discussion]], additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

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

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

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~Q_{n - \frac{1}{2}}^{1}(z)</math>		<math>Q_{n - \frac{1}{2}}^{1}(z)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(n-\tfrac{1}{2}) (z^2-1)^{-\frac{1}{2}} [z Q_{n - \frac{1}{2}}(z) - Q_{n - \frac{3}{2}}(z)]		(n-\tfrac{1}{2}) (z^2-1)^{-\frac{1}{2}} [z Q_{n - \frac{1}{2}}(z) - Q_{n - \frac{3}{2}}(z)]
	</math>		</math>
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	<td allign="center">    …    </td>		<td allign="center">    …    </td>
	<td align="left">		<td align="left">
	for <math>~n \ge 1 \, ,</math>		for <math>n \ge 1 \, ,</math>
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~Q_{n - \frac{1}{2}}^{2}(z)</math>		<math>Q_{n - \frac{1}{2}}^{2}(z)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~		<math>
	(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}		(z^2-1)^{-\frac{1}{2}} \{ (n-\tfrac{3}{2}) z Q^1_{n - \frac{1}{2}}(z) - (n+\tfrac{1}{2})Q^1_{n - \frac{3}{2}}(z)\}
	</math>		</math>
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	<td allign="center">    …    </td>		<td allign="center">    …    </td>
	<td align="left">		<td align="left">
	for <math>~n \ge 1 \, .</math>		for <math>n \ge 1 \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	As an example, the first of these two relations has been used to generate a column of numbers in the [[#Comparison_with_Table_IX_from_MF53\|comparison table shown below]] for the toroidal function, <math>~Q^1_{+\frac{1}{2}}(z)</math>.		As an example, the first of these two relations has been used to generate a column of numbers in the [[#Comparison_with_Table_IX_from_MF53\|comparison table shown below]] for the toroidal function, <math>Q^1_{+\frac{1}{2}}(z)</math>.

	====Comparison with Table IX from MF53====		====Comparison with Table IX from MF53====

Jet53man: /* Example Recurrence Relations */

2021-07-18T20:11:37Z

Example Recurrence Relations

← Older revision		Revision as of 16:11, 18 July 2021
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	The above [[#Analytic_Expressions_.26_Plots\|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>~P^0_{-\frac{1}{2}}(z)</math> and <math>~P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>~Q^0_{-\frac{1}{2}}(z)</math> and <math>~Q^0_{+\frac{1}{2}}(z)</math>. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"		The above [[#Analytic_Expressions_.26_Plots\|''Toroidal Function Evaluations'']] table provides analytic expressions for the pair of foundation functions, <math>~P^0_{-\frac{1}{2}}(z)</math> and <math>~P^0_{+\frac{1}{2}}(z)</math>, and the associated pair of foundation functions, <math>~Q^0_{-\frac{1}{2}}(z)</math> and <math>~Q^0_{+\frac{1}{2}}(z)</math>. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

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

	Specifically, letting <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,		Specifically, letting <math>~\mu \rightarrow 0</math> and <math>~\nu \rightarrow (m - \tfrac{1}{2})</math>, for all <math>~m \ge 2</math>, we have,

Jet53man: /* Caption for Plots */

2021-07-18T20:09:26Z

Caption for Plots

← Older revision		Revision as of 16:09, 18 July 2021
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	* The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>];		* The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>];
	* The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>K(k)</math> and <math>E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>]. (See an accompanying sample of [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|elliptic integral values extracted]] from [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>].)		* The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>K(k)</math> and <math>E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>]. (See an accompanying sample of [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|elliptic integral values extracted]] from [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>].)
	* The dashed red curve was also derived using formulae expressed in terms of the complete elliptic integrals, but the ''values'' of the elliptic integrals have been calculated using (double-precision versions of) algorithms drawn from ''Numerical Recipes.''		* The dashed red curve was also derived using formulae expressed in terms of the complete elliptic integrals, but the ''values'' of the elliptic integrals have been calculated using (double-precision versions of) algorithms drawn from [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''].


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	ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [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)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>. As can be seen in the [[#Comparison_with_Table_IX_from_MF53\|table of function values immediately below]], this is entirely consistent with the lower-precision value that we have extracted from [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], and exactly matches the double-precision value we have calculated based on the ''Numerical Recipes'' algorithm.		ADDITIONAL NOTE:   In ''Example 4'' on p. 340 of [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)], we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of <math>~Q^0_{-\frac{1}{2}}(z = 2.6) = 1.419337751</math>. As can be seen in the [[#Comparison_with_Table_IX_from_MF53\|table of function values immediately below]], this is entirely consistent with the lower-precision value that we have extracted from [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], and exactly matches the double-precision value we have calculated based on the [https://www.amazon.com/Numerical-Recipes-Fortran-Scientific-Computing/dp/052143064X ''Numerical Recipes''] algorithm.
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Caption for Plots */

2021-07-18T20:06:17Z

Caption for Plots

← Older revision		Revision as of 16:06, 18 July 2021
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	For each choice of the integer indexes, <math>n \ge 0</math> and <math>m \ge 0</math>, the relevant plot shows how the function, <math>X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>z</math>. In each plot …		For each choice of the integer indexes, <math>n \ge 0</math> and <math>m \ge 0</math>, the relevant plot shows how the function, <math>X^n_{m-\frac{1}{2}}(z)</math>, varies with <math>z</math>. (Click on the small plot image to view an enlarged image.) In each plot …
	* The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>];		* The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>];
	* The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>K(k)</math> and <math>E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>]. (See an accompanying sample of [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|elliptic integral values extracted]] from [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>].)		* The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, <math>K(k)</math> and <math>E(k)</math>, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>]. (See an accompanying sample of [[2DStructure/ToroidalCoordinateIntegrationLimits#Evaluation_of_Elliptic_Integrals\|elliptic integral values extracted]] from [<b>[[Appendix/References#CRC\|<font color="red">CRC</font>]]</b>].)
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	NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.		NOTE: The tabulated values of the function, <math>~Q^1_{-\frac{1}{2}}</math>, that appear in Table IX (p. 1923) of [<b>[[User:Tohline/Appendix/References#MF53\|<font color="red">MF53</font>]]</b>] — also see [[#Comparison_with_Table_IX_from_MF53\|immediately below]] — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this ''specific'' function — both here and in our [[User:Tohline/Appendix/Mathematics/ToroidalSynopsis01#Q1Q2Summary\|accompanying discussion]] — we have plotted the absolute value of the function, <math>~\|Q^1_{-\frac{1}{2}}(z)\|</math>.

Jet53man: /* Comparison with Table IX from MF53 */

2021-07-18T01:59:17Z

Comparison with Table IX from MF53

← Older revision		Revision as of 21:59, 17 July 2021
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	<div align="center" id="TabulatedValues">		<div align="center" id="TabulatedValues">
	<table border="1" cellpadding="8" width="50%" align="center">		<table border="1" cellpadding="8" width="90%" align="center">
	<tr>		<tr>
	<td align="center">		<td align="center">
	Top half of Table IX (p. 1923) of [<b>[[~~User:Tohline/~~Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]		Top half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="center">		<td align="center">
	Bottom half of Table IX (p. 1923) of [<b>[[~~User:Tohline/~~Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]		Bottom half of Table IX (p. 1923) of [<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>]
	</td>		</td>
	</tr>		</tr>