User:Tohline/2DStructure/ToroidalGreenFunction - Revision history

Tohline: /* Summary Table */

2020-07-15T22:48:03Z

Summary Table

← Older revision		Revision as of 22:48, 15 July 2020
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	=Appendix B: Elliptic Integrals=		=Appendix B: Elliptic Integrals=
	==Summary Table==		==Summary Table==
	If the default argument of a 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 [[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:

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

Tohline: /* Series Expansions */

2020-07-15T22:46:17Z

Series Expansions

← Older revision		Revision as of 22:46, 15 July 2020
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	==Series Expansions==		==Series Expansions==

	~~In §8.113 and §8.114 of~~ [~~http~~://~~www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)~~] we find the following series expansions for complete elliptic integrals of, respectively, the first and second kind:		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:
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\frac{2K(k)}{\pi}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~1 + \biggl( \frac{1}{2} \biggr)^2k^2~~
	~~+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4~~
	~~+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6~~
	~~+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8~~
	~~+ \cdots~~
	~~+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}~~
	~~+ \cdots~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\frac{2E(k)}{\pi}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~1 - \frac{1}{2^2} ~k^2~~
	~~- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4~~
	~~- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}~~
	~~- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}~~
	~~~-~ \cdots -~~
	~~\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}~~
	~~~-~ \cdots~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~

			<div align="center">
			{{ User:Tohline/Math/EQ_EllipticIntegral01 }}<br />
			{{ User:Tohline/Math/EQ_EllipticIntegral02 }}
			</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].

Tohline: /* Series Expansions */

2020-07-10T19:50:50Z

Series Expansions

← Older revision		Revision as of 19:50, 10 July 2020
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	+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4		+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
	+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6		+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
			+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8
	+ \cdots		+ \cdots
	+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}		+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
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	- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4		- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
	- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}		- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
	~-~ \cdots		- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7}
			~-~ \cdots -
	\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}		\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
	~-~ \cdots		~-~ \cdots

Tohline: /* Series Expansions */

2020-07-10T18:19:21Z

Series Expansions

← Older revision		Revision as of 18:19, 10 July 2020
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	</tr>		</tr>
	</table>		</table>

			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].

	=See Also=		=See Also=

Tohline: /* Summary Table */

2018-10-11T20:34:17Z

Summary Table

← Older revision		Revision as of 20:34, 11 October 2018
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~</math>		<math>~E(k)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~</math>		<math>~=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~</math>		<math>~(1+k^\prime)E(k_1) - k^\prime K(k)</math>
	</td>		</td>
			</tr>

			<tr>
			<td align="center" colspan="3">[http://dlmf.nist.gov/19.8.E12 DLMS §19.8.12]</td>
	</tr>		</tr>
	</table>		</table>
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	<math>~(1+k_1)K(k_1)</math>		<math>~(1+k_1)K(k_1)</math>
	</td>		</td>
			</tr>

			<tr>
			<td align="center" colspan="3">[http://dlmf.nist.gov/19.8.E12 DLMS §19.8.12]</td>
	</tr>		</tr>

Tohline: /* Summary Table */

2018-10-11T20:19:04Z

Summary Table

← Older revision		Revision as of 20:19, 11 October 2018
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	</tr>		</tr>
	<tr>		<tr>
	<td align="center" width="20%"><math>~k</math></td>		<td align="center" width="20%"><math>~k</math><br />
	<td align="center" width="20%"><math>~k_1</math></td>		<font size="-1">([http://dlmf.nist.gov/22.7.i Descending Landen Transformation])</font>
			</td>
			<td align="center" width="20%"><math>~k_1</math><br />
			<font size="-1">([http://dlmf.nist.gov/22.7.ii Ascending Landen Transformation])</font>
			</td>
	<td align="center" width="20%><math>~\frac{R}{R_1} = e^{-\eta}</math></td>		<td align="center" width="20%><math>~\frac{R}{R_1} = e^{-\eta}</math></td>
	<td align="center"><math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr)</math></td>		<td align="center"><math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr)</math></td>

Tohline: /* Series Expansions */

2018-09-30T00:11:45Z

Series Expansions

← Older revision		Revision as of 00:11, 30 September 2018
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	<td align="left">		<td align="left">
	<math>~		<math>~
	1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4		1 + \biggl( \frac{1}{2} \biggr)^2k^2
			+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
			+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6
	+ \cdots		+ \cdots
	+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}		+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
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	</td>		</td>
	</tr>		</tr>
	~~<table border="0" cellpadding="5" align="center">~~

	<tr>		<tr>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	1 - \frac{1}{2^2} ~k^2 - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4		1 - \frac{1}{2^2} ~k^2
	- \cdots		- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
			- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
			~-~ \cdots
	\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}		\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
	- \cdots		~-~ \cdots
	</math>		</math>
	</td>		</td>

Tohline: /* Series Expansions */

2018-09-29T23:54:40Z

Series Expansions

← Older revision		Revision as of 23:54, 29 September 2018
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	==Series Expansions==		==Series Expansions==

	In §8.113 of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], we find the following series expansions for complete elliptic integrals of the first and second kind:		In §8.113 and §8.114 of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)] we find the following series expansions for complete elliptic integrals of, respectively, the first and second kind:
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>~		<math>~
	1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \cdots		1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4
			+ \cdots
			+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
			+ \cdots
			</math>
			</td>
			</tr>
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{2E(k)}{\pi}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1 - \frac{1}{2^2} ~k^2 - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4
			- \cdots
			\biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
			- \cdots
	</math>		</math>
	</td>		</td>

Tohline: /* Appendix B: Elliptic Integrals */

2018-09-29T23:45:30Z

Appendix B: Elliptic Integrals

← Older revision		Revision as of 23:45, 29 September 2018
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	'''Q.E.D.'''		'''Q.E.D.'''

			==Series Expansions==

			In §8.113 of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], we find the following series expansions for complete elliptic integrals of the first and second kind:
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{2K(k)}{\pi}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \cdots
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Summary Table */

2018-09-28T21:33:41Z

Summary Table

← Older revision		Revision as of 21:33, 28 September 2018
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	As published, for example, in §8.126 of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], the following mappings are valid:		<span id="GandR65">As published, for example,</span> in §8.126 of [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], the following mappings are valid:

	<table border="1" cellpadding="8" width="85%" align="center">		<table border="1" cellpadding="8" width="85%" align="center">