Appendix/Ramblings/PowerSeriesExpressions - Revision history

Jet53man: /* Broadly Used Mathematical Expressions (shown here without proof) */

2024-06-28T23:46:57Z

Broadly Used Mathematical Expressions (shown here without proof)

← Older revision		Revision as of 19:46, 28 June 2024
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	</table>		</table>
	</div>		</div>


	==Expressions with Astrophysical Relevance==		==Expressions with Astrophysical Relevance==

Jet53man: /* Isothermal Lane-Emden Function */

2022-04-18T22:44:15Z

Isothermal Lane-Emden Function

← Older revision		Revision as of 18:44, 18 April 2022
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	===Isothermal Lane-Emden Function===		===Isothermal Lane-Emden Function===

	<!-- As we have discussed in [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Governing_Relations\|a separate chapter]], the 2<sup>nd</sup>-order ODE that governs the radial density distribution in an isothermal sphere is,		<!-- As we have discussed in [[SSC/Structure/IsothermalSphere#Governing_Relations\|a separate chapter]], the 2<sup>nd</sup>-order ODE that governs the radial density distribution in an isothermal sphere is,
	<div align="center" id="Chandrasekhar">		<div align="center" id="Chandrasekhar">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	-->		-->

	Here we seek a power-series expression for the isothermal, Lane-Emden function — expanded about the coordinate center — that approximately satisfies the [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Chandrasekhar\|isothermal Lane-Emden equation]]; making the variable substitution (sorry for the unnecessary complication!), <math>~\psi(\xi) \leftrightarrow w(r)</math>, the governing ODE is,		Here we seek a power-series expression for the isothermal, Lane-Emden function — expanded about the coordinate center — that approximately satisfies the [[SSC/Structure/IsothermalSphere#Chandrasekhar\|isothermal Lane-Emden equation]]; making the variable substitution (sorry for the unnecessary complication!), <math>~\psi(\xi) \leftrightarrow w(r)</math>, the governing ODE is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	See also:		See also:
	* Equation (377) from §22 in Chapter IV of [[~~User:Tohline/~~Appendix/References#C67\|C67]]).		* Equation (377) from §22 in Chapter IV of [[Appendix/References#C67\|C67]].

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-22T23:48:14Z

Infinitesimally Thin Axisymmetric Disk

← Older revision		Revision as of 19:48, 22 March 2022
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	\biggr]\biggl(\frac{c}{a}\biggr)^8		\biggr]\biggl(\frac{c}{a}\biggr)^8
	+		+
	\pi \biggl[\frac{5 \cdot 7 ~~\cdot 9~~}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9		\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
	</math>		</math>
	</td>		</td>
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	\biggr]\biggl(\frac{c}{a}\biggr)^8		\biggr]\biggl(\frac{c}{a}\biggr)^8
	+		+
	\pi \biggl[\frac{5 \cdot 7 ~~\cdot 9~~}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9		\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
	+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .		+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
	</math>		</math>
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	\biggr]\biggl(\frac{c}{a}\biggr)^8		\biggr]\biggl(\frac{c}{a}\biggr)^8
	+		+
	\pi \biggl[\frac{5 \cdot 7 ~~\cdot 9~~}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9		\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
	\biggr\}		\biggr\}
	+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)		+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
Line 3,048:		Line 3,048:
	\biggr]\biggl(\frac{c}{a}\biggr)^8		\biggr]\biggl(\frac{c}{a}\biggr)^8
	-		-
	\pi \biggl[\frac{5 \cdot 7 ~~\cdot 9~~}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9		\pi \biggl[\frac{3^2\cdot 5 \cdot 7}{2^8}\biggr]\biggl(\frac{c}{a}\biggr)^9
	\biggr\}		\biggr\}
	+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)\, .		+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)\, .

Jet53man: /* Nearly Spherical Configurations */

2022-03-22T22:53:59Z

Nearly Spherical Configurations

← Older revision		Revision as of 18:53, 22 March 2022
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	+ e^4 \biggl[\frac{3^2 - 2\cdot 5 - 3\cdot 5}{2^3 \cdot 3 \cdot 5}\biggr]		+ e^4 \biggl[\frac{3^2 - 2\cdot 5 - 3\cdot 5}{2^3 \cdot 3 \cdot 5}\biggr]
	+ e^6 \biggl[ \frac{ 3\cdot 5^2 - 3^2\cdot 7 - 5\cdot 7 - 3\cdot 5\cdot 7}{2^4 \cdot 3 \cdot 5 \cdot 7}\biggr]		+ e^6 \biggl[ \frac{ 3\cdot 5^2 - 3^2\cdot 7 - 5\cdot 7 - 3\cdot 5\cdot 7}{2^4 \cdot 3 \cdot 5 \cdot 7}\biggr]
	+ e^8 \biggl[\frac{ ~~3\cdot~~ 5^2 \cdot 7^2 - 2^2 \cdot 3^2\cdot 5^2 - 2\cdot 3^3 \cdot 7 - 2^2\cdot 3\cdot 5\cdot 7		+ e^8 \biggl[\frac{ 5^2 \cdot 7^2 - 2^2 \cdot 3^2\cdot 5^2 - 2\cdot 3^3 \cdot 7 - 2^2\cdot 3\cdot 5\cdot 7
	- 3^2 \cdot 5^2 \cdot 7 }{2^7 \cdot 3^2 \cdot 5 \cdot 7} \biggr]		- 3^2 \cdot 5^2 \cdot 7 }{2^7 \cdot 3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{10})		+ \mathcal{O}(e^{10})
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	- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]		- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
	- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]		- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
	+ e^8 \biggl[\frac{ ~~201~~ }{~~2^6 \cdot~~ 3^2 \cdot 5 \cdot 7} \biggr]		- e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{10})		+ \mathcal{O}(e^{10})
	</math>		</math>
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	- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]		- e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
	- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]		- e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
	+ e^8 \biggl[\frac{ ~~201~~ }{~~2^6 \cdot~~ 3^2 \cdot 5 \cdot 7} \biggr]		- e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{10})		+ \mathcal{O}(e^{10})
	</math>		</math>
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	- e^2 \biggl[\frac{2}{3 \cdot 5}\biggr]		- e^2 \biggl[\frac{2}{3 \cdot 5}\biggr]
	- e^4 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]		- e^4 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
	+ e^6 \biggl[\frac{ ~~201~~ }{~~2^6 \cdot~~ 3^2 \cdot 5 \cdot 7} \biggr]		- e^6 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{8}) \, .		+ \mathcal{O}(e^{8}) \, .
	</math>		</math>
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	+ e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]		+ e^4 \biggl[\frac{2}{3 \cdot 5}\biggr]
	+ e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]		+ e^6 \biggl[ \frac{ 2^3}{3 \cdot 5 \cdot 7}\biggr]
	- e^8 \biggl[\frac{ ~~201~~ }{~~2^6 \cdot~~ 3^2 \cdot 5 \cdot 7} \biggr]		+ e^8 \biggl[\frac{ 2^4 }{3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{10})		+ \mathcal{O}(e^{10})
	</math>		</math>
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	+ e^2 \biggl[\frac{2^2}{3 \cdot 5}\biggr]		+ e^2 \biggl[\frac{2^2}{3 \cdot 5}\biggr]
	+ e^4 \biggl[ \frac{ 2^4}{3 \cdot 5 \cdot 7}\biggr]		+ e^4 \biggl[ \frac{ 2^4}{3 \cdot 5 \cdot 7}\biggr]
	- e^6 \biggl[\frac~~{ 201 }~~{2^5 ~~\cdot~~ 3^2 \cdot 5 \cdot 7} \biggr]		+ e^6 \biggl[\frac{ 2^5 }{3^2 \cdot 5 \cdot 7} \biggr]
	+ \mathcal{O}(e^{8}) \, .		+ \mathcal{O}(e^{8}) \, .
	</math>		</math>

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-22T16:56:35Z

Infinitesimally Thin Axisymmetric Disk

@@ Line 2,557: / Line 2,557: @@
 - \biggl[ \frac{2^4}{5 \cdot 7}\biggr] \biggl(\frac{c}{a}\biggr)^7
-<tr>
+- \biggl[ \frac{2^7}{3^2 \cdot 5 \cdot 7}\biggr] \biggl(\frac{c}{a}\biggr)^9
-  <td align="right">
++ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \,  .
 </math>
    </td>

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-22T16:41:01Z

Infinitesimally Thin Axisymmetric Disk

@@ Line 2,386: / Line 2,386: @@
 </table>
-(continue reduction)
+(continue expression simplification)
 <table border="0" cellpadding="5" align="center">
@@ Line 2,501: / Line 2,501: @@
 + \biggl[\frac{5}{2^4 }\biggr]
 \biggr\}\biggl(\frac{c}{a}\biggr)^7
 </math>
    </td>

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-21T23:58:30Z

Infinitesimally Thin Axisymmetric Disk

← Older revision		Revision as of 19:58, 21 March 2022
Line 2,380:		Line 2,380:

	- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9		- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9
	+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)		+ \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr) \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>

			(continue reduction)

			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	~~ ~~		<math>\frac{\sin^{-1}e}{e} = \cos^{-1}\biggl(\frac{c}{a}\biggr) \biggl[ 1 - \frac{c^2}{a^2} \biggr]^{-1 / 2}</math>
	</td>		</td>
	<td align="center">		<td align="center">

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-21T23:54:50Z

Infinitesimally Thin Axisymmetric Disk

← Older revision		Revision as of 19:54, 21 March 2022
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	<td align="left">		<td align="left">
	<math>		<math>
	- \biggl(\frac{c}{a}\biggr) - \biggl[\~~frac{1}~~{~~2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^3~~		- \biggl(\frac{c}{a}\biggr)
	- \biggl[\frac{3 }{2^3\cdot 5}\biggr]~~\biggl(\frac{c}{a}\biggr)^5~~		- \biggl\{
	~~- \biggl[~~ \frac{5}{2~~^4\cdot 7~~}\biggr~~]\biggl(\frac{c}{a}\biggr)^7~~		\biggl[\frac{1}{2\cdot 3}\biggr]
	~~</math>~~		+ \frac{1}{2}
	~~</td>~~		\biggr\}\biggl(\frac{c}{a}\biggr)^3
	~~</tr>~~		- \biggl\{
			\biggl[\frac{3 }{2^3\cdot 5}\biggr]
	~~<tr>~~		+ \biggl[\frac{1}{2^2\cdot 3}\biggr]
	~~<td align="right">~~		+ \biggl[\frac{3}{2^3 }\biggr]
	~~ ~~		\biggr\}\biggl(\frac{c}{a}\biggr)^5
	~~</td>~~
	~~<td align="center">~~
	~~ ~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~
	- \~~frac{1}{2~~} \biggl(\frac{c}{a}\biggr)^3

	- \biggl[\~~frac~~{~~1}{2^2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5~~
	- \biggl[\frac{3}{2^3 }\biggr] ~~\biggl(\frac{c}{a}\biggr)^5~~

	- \biggl[\frac{ 3 }{2^4 \cdot 5}\biggr]~~\biggl(\frac{c}{a}\biggr)^7~~
	- \biggl[\frac{1}{2^4}\biggr]~~\biggl(\frac{c}{a}~~\biggr~~)^7~~
	~~- \biggl[~~\~~frac{5~~}~~{2^4 }\biggr]~~ \biggl(\frac{c}{a}\biggr)^7

			- \biggl\{
			\biggl[ \frac{5}{2^4\cdot 7}\biggr]
			+ \biggl[\frac{ 3 }{2^4 \cdot 5}\biggr]
			+ \biggl[\frac{1}{2^4}\biggr]\
			+ \biggl[\frac{5}{2^4 }\biggr]
			\biggr\}\biggl(\frac{c}{a}\biggr)^7
	</math>		</math>
	</td>		</td>

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-21T23:10:06Z

Infinitesimally Thin Axisymmetric Disk

@@ Line 2,449: / Line 2,449: @@
 - \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9
 + \mathcal{O}\biggl(\frac{c^{10}}{a^{10}}\biggr)
 </math>

Jet53man: /* Infinitesimally Thin Axisymmetric Disk */

2022-03-21T23:01:51Z

Infinitesimally Thin Axisymmetric Disk

← Older revision		Revision as of 19:01, 21 March 2022
Line 2,428:		Line 2,428:
	<math>		<math>
	- \frac{1}{2} \biggl(\frac{c}{a}\biggr)^3 - \biggl[\frac{1}{2^2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5		- \frac{1}{2} \biggl(\frac{c}{a}\biggr)^3 - \biggl[\frac{1}{2^2\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^5
	- \biggl[\frac{~~1\cdot~~ 3 }{2^~~2\cdot~~ 4\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^7		- \biggl[\frac{ 3 }{2^4 \cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^7
	- \biggl[ \frac{~~1 \cdot 3\cdot~~ 5}{2^~~2\cdot 4\cdot 6~~\cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^9		- \biggl[ \frac{5}{2^5 \cdot 7}\biggr]\biggl(\frac{c}{a}\biggr)^9
	</math>		</math>
	</td>		</td>
Line 2,443:		Line 2,443:
	<td align="left">		<td align="left">
	<math>		<math>
	- \biggl[\frac{3}{2^3 }\biggr] \biggl(\frac{c}{a}\biggr)^5 - \biggl[\frac{3}{2^~~3 }\biggr] \biggl[\frac{1}{2\cdot 3~~}\biggr]\biggl(\frac{c}{a}\biggr)^7		- \biggl[\frac{3}{2^3 }\biggr] \biggl(\frac{c}{a}\biggr)^5 - \biggl[\frac{1}{2^4}\biggr]\biggl(\frac{c}{a}\biggr)^7
	- \biggl[\frac{3}{2^~~3 }\biggr] \biggl[\frac{1\cdot 3 }{2\cdot 4~~\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^9		- \biggl[\frac{3^2 }{2^6\cdot 5}\biggr]\biggl(\frac{c}{a}\biggr)^9

	- \biggl[\frac{5}{2^4 }\biggr] \biggl(\frac{c}{a}\biggr)^7 - \biggl[\frac{5}{2^~~4 }\biggr] \biggl[\frac{1}{2~~\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^9		- \biggl[\frac{5}{2^4 }\biggr] \biggl(\frac{c}{a}\biggr)^7 - \biggl[\frac{5}{2^5\cdot 3}\biggr]\biggl(\frac{c}{a}\biggr)^9

	- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9		- \biggl[\frac{ 5\cdot 7}{2^7 }\biggr]\biggl(\frac{c}{a}\biggr)^9