Appendix/Ramblings/PowerSeriesExpressions - Revision history

Joel2: /* Power-Series Derivation */

2026-10-15T22:44:11Z

Power-Series Derivation

← Older revision		Revision as of 17:44, 15 October 2026
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	</td></tr></table>		</td></tr></table>
	</div>		</div>


			See also:
			* Equation (62) from §5 in Chapter IV of [[Appendix/References#C67\|C67]].


	NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, the analogous power-series expression appears as equation (15) in the article by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.		NOTE:  For cylindrically symmetric, rather than spherically symmetric, configurations, the analogous power-series expression appears as equation (15) in the article by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-15T22:33:42Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 17:33, 15 October 2026
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	<td align="left">		<td align="left">
	<math>		<math>
	1 - \frac{\xi^2}{30} + \frac{\xi^4}{1260} + \cdots		1 - \frac{\xi^2}{30} + \frac{\xi^4}{1260} + \cdots ~~~~~\Rightarrow ~~ \frac{2}{3} x_Y = \frac{2}{3} - \frac{\xi^2}{45} + \frac{\xi^4}{1890} + \cdots
	</math>		</math>
	</td>		</td>
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	+ \biggl[\frac{\xi^6}{900} \biggr]		+ \biggl[\frac{\xi^6}{900} \biggr]
	</math>   (<font color="orange">-0.333587685</font>, or <font color="green">-0.3356304</font>)		</math>   (<font color="orange">-0.333587685</font>, or <font color="green">-0.3356304</font>)
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>\Rightarrow ~~~ x_Y</math>
			</td>
			<td align="center">
			<math>\approx</math>
			</td>
			<td align="left">
			<math>
			\frac{2}{3} - \biggl[\frac{\xi^2}{45} \biggr]
			+ \biggl[ \frac{\xi^4}{1890} \biggr]
			+ \biggl[\frac{\xi^6}{900} \biggr]
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			<b><font color="red">YES!!</font></b>
	</td></tr></table>		</td></tr></table>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-15T21:31:18Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 16:31, 15 October 2026
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	<td align="left">		<td align="left">
	<math>		<math>
	\overbrace{\biggl[- \frac{1}{3} + \frac{\xi^2}{2\cdot 3\cdot 5} - \frac{\xi^4}{3^2 \cdot 5\cdot 7} + \frac{61 \xi^6}{ 2^3 \cdot 3^6\cdot 5\cdot 7}\biggr]}^{-0.33295212}		\overbrace{\biggl[- \frac{1}{3} + \frac{\xi^2}{2\cdot 3\cdot 5} - \frac{\xi^4}{3^2 \cdot 5\cdot 7} + \frac{61 \xi^6}{ 2^3 \cdot 3^6\cdot 5\cdot 7}\biggr]}^{-0.33295212 ~~ or ~~ - 0.329910995}
	\times \underbrace{\biggl\{		\times \underbrace{\biggl\{
	1 + \frac{\xi^2}{6}		1 + \frac{\xi^2}{6}
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	\frac{1}{2^2\cdot 3^4\cdot 5\cdot 7}		\frac{1}{2^2\cdot 3^4\cdot 5\cdot 7}
	\biggr]		\biggr]
	\biggr\}}_{1.001908167}		\biggr\}}_{1.001908167 ~ or ~ 1.017340}
	\approx </math>   <font color="orange">-0.333587444</font>		\approx </math>   <font color="orange">-0.333587444</font>, or <font color="green">-0.3356316</font>
	</td>		</td>
	</tr>		</tr>
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	+ \biggl[ \frac{\xi^4}{1890} \biggr]		+ \biggl[ \frac{\xi^4}{1890} \biggr]
	+ \biggl[\frac{\xi^6}{900} \biggr]		+ \biggl[\frac{\xi^6}{900} \biggr]
	</math>   <font color="orange">(-0.333587685)</font>		</math>   (<font color="orange">-0.333587685</font>, or <font color="green">-0.3356304</font>)
	</td>		</td>
	</tr>		</tr>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-15T21:15:48Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 16:15, 15 October 2026
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	1 + \psi + \frac{\psi^2}{2!} + \frac{\psi^3}{3!} + \frac{\psi^4}{4!} + \cdots		1 + \psi + \frac{\psi^2}{2!} + \frac{\psi^3}{3!} + \frac{\psi^4}{4!} + \cdots
	</math>      (1 + 5.227213 × 10<sup>-4</sup>		</math>      (1 + 5.227213 × 10<sup>-4</sup>
	, or <font color="orange">1+1.908894 × 10<sup>-03</sup></font>		, or <font color="orange">1+1.908894 × 10<sup>-03</sup></font>, or <font color="green">1.017340</font>
	vs. from above   1.000522 × 10<sup>0</sup>  , or <font color="orange">1+1.908895 × 10<sup>-03</sup></font>)		vs. from above   1.000522 × 10<sup>0</sup>  , or <font color="orange">1+1.908895 × 10<sup>-03</sup></font>, or <font color="green">1.017340293</font>)
	</td>		</td>
	</tr>		</tr>
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	\frac{-4}{45360}		\frac{-4}{45360}
	\biggr]		\biggr]
	</math>      (1 + 5.22721 × 10<sup>-4</sup>, or <font color="orange">1 + 1.908894 × 10<sup>-03</sup></font>)		</math>      (1 + 5.22721 × 10<sup>-4</sup>, or <font color="orange">1 + 1.908894 × 10<sup>-03</sup></font>, or <font color="green">1.017340</font>)
	</td>		</td>
	</tr>		</tr>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-15T20:58:49Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 15:58, 15 October 2026
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	\frac{\xi^8}{2^5\cdot 3^2 \cdot 5}		\frac{\xi^8}{2^5\cdot 3^2 \cdot 5}
	+\mathcal{O}(\xi^{10})		+\mathcal{O}(\xi^{10})
	</math>      (1.42715 × 10<sup>-10</sup>  , or <font color="orange">6.935893 × 10<sup>-09</sup></font>		</math>      (1.42715 × 10<sup>-10</sup>  , or <font color="orange">6.935893 × 10<sup>-09</sup></font>, or <font color="green">5.080120 × 10<sup>-06</sup></font>
	vs. from above  1.42715 × 10<sup>-10</sup>  , or <font color="orange">6.935909 × 10<sup>-09</sup></font> )		vs. from above  1.42715 × 10<sup>-10</sup>  , or <font color="orange">6.935909 × 10<sup>-09</sup></font>, or <font color="green">5.081054 × 10<sup>-06</sup></font> )
	</td>		</td>
	</tr>		</tr>
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	\frac{\xi^8}{2^4 \cdot 3^4}		\frac{\xi^8}{2^4 \cdot 3^4}
	+\mathcal{O}(\xi^{10})		+\mathcal{O}(\xi^{10})
	</math>      (7.46276 × 10<sup>-14</sup>  , or <font color="orange">1.325760 × 10<sup>-11</sup></font>		</math>      (7.46276 × 10<sup>-14</sup>  , or <font color="orange">1.325760 × 10<sup>-11</sup></font>, or <font color="green">8.917476 × 10<sup>-08</sup></font>
	vs. from above  7.45808 × 10<sup>-14</sup>  , or <font color="orange">1.322730 × 10<sup>-11</sup></font> )		vs. from above  7.45808 × 10<sup>-14</sup>  , or <font color="orange">1.322730 × 10<sup>-11</sup></font>, or <font color="green">8.735179 × 10<sup>-08</sup></font>)
	</td>		</td>
	</tr>		</tr>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-15T19:00:45Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 14:00, 15 October 2026
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	</td></tr></table>		</td></tr></table>
	</div>		</div>


	<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">		<table border="1" align="center" width="80%" cellpadding="5"><tr><td align="left">
	<font color="red">		<font color="red">
	This is a derivation check ...</font>      (Numerical values in parentheses assume <math>\xi = 0.056</math>, or <font color="orange">0.107</font>)		This is a derivation check ...</font>      (Numerical values in parentheses assume <math>\xi = 0.056</math>, or <font color="orange">0.107</font>, or <font color="green">0.322</font>)
	</font>		</font>

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	<math>		<math>
	\frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} + \cdots		\frac{\xi^2}{6} - \frac{\xi^4}{120} + \frac{\xi^6}{1890} - \frac{61 \xi^8}{1,632,960} + \cdots
	</math>      (5.22585 × 10<sup>-4</sup>, or <font color="orange">1.907075 × 10<sup>-03</sup></font>)<math>~~~\Rightarrow ~~~ e^\psi =</math>(1.000522, or <font color="orange">1.001908895</font>)		</math>      (5.22585 × 10<sup>-4</sup>, or <font color="orange">1.907075 × 10<sup>-03</sup></font>, or <font color="green">0.017191666</font>)<math>~~~\Rightarrow ~~~ e^\psi =</math>(1.000522, or <font color="orange">1.001908895</font>, or <font color="green">1.017340293</font>)
	</td>		</td>
	</tr>		</tr>
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	<math>		<math>
	\frac{1}{3} - \frac{\xi^2}{30} + \frac{6\xi^4}{1890} - \frac{8\cdot 61 \xi^6}{1,632,960} + \cdots		\frac{1}{3} - \frac{\xi^2}{30} + \frac{6\xi^4}{1890} - \frac{8\cdot 61 \xi^6}{1,632,960} + \cdots
	</math>      (3.33229× 10<sup>-1</sup>, or <font color="orange">0.332952116</font>)		</math>      (3.33229× 10<sup>-1</sup>, or <font color="orange">0.332952116</font>, or <font color="green">0.329910995</font>)
	<math>~~~\Rightarrow ~~~ x_Y = </math>(1 -  0.465010   =   0.34990)		<math>~~~\Rightarrow ~~~ x_Y = </math>(1 -  0.465010   =   0.34990)
	<math>(1-x_Y) = </math> (xxx, or <font color="orange">0.333587080</font>)		<math>(1-x_Y) = </math> (xxx, or <font color="orange">0.333587080</font>, or <font color="green">0.335631748</font>)
	</td>		</td>
	</tr>		</tr>
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	+ \frac{\xi^8}{2^6\cdot 3^4 \cdot 5^2 \cdot 7}\biggl[2^5\cdot 5 + 3^2\cdot 7 \biggr]		+ \frac{\xi^8}{2^6\cdot 3^4 \cdot 5^2 \cdot 7}\biggl[2^5\cdot 5 + 3^2\cdot 7 \biggr]
	+\mathcal{O}(\xi^{10})		+\mathcal{O}(\xi^{10})
	</math>      (2.73095 × 10<sup>-07</sup>, or <font color="orange">3.636931 × 10<sup>-06</sup></font>		</math>      (2.73095 × 10<sup>-07</sup>, or <font color="orange">3.636931 × 10<sup>-06</sup></font>, or <font color="green">2.955536 × 10<sup>-04</sup></font>
	vs. from above  2.73095 × 10<sup>-7</sup>, or <font color="orange">3.636935 × 10<sup>-06</sup></font> )		vs. from above  2.73095 × 10<sup>-7</sup>, or <font color="orange">3.636935 × 10<sup>-06</sup></font> , or <font color="green">2.955533 × 10<sup>-04</sup></font>)
	</td>		</td>
	</tr>		</tr>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-13T23:28:30Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 18:28, 13 October 2026
Line 3,056:		Line 3,056:
	\frac{1}{2^2\cdot 3^4\cdot 5\cdot 7}		\frac{1}{2^2\cdot 3^4\cdot 5\cdot 7}
	\biggr]		\biggr]
	\biggr\}}_{1.~~001908894~~}		\biggr\}}_{1.001908167}
	\approx </math>   <font color="orange">-0.~~333586729~~</font>		\approx </math>   <font color="orange">-0.333587444</font>
	</td>		</td>
	</tr>		</tr>
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	+ \biggl[ \frac{\xi^4}{1890} \biggr]		+ \biggl[ \frac{\xi^4}{1890} \biggr]
	+ \biggl[\frac{\xi^6}{900} \biggr]		+ \biggl[\frac{\xi^6}{900} \biggr]
	\, .		</math>   <font color="orange">(-0.333587685)</font>
	</~~math~~>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-13T23:14:17Z

Displacement Function for Isothermal LAWE

@@ Line 3,226: / Line 3,226: @@
 - \frac{1}{3} - \xi^2\biggl[\frac{1}{3^2 \cdot 5} \biggr]
 + \xi^4\biggl[ \frac{1}{2 \cdot 3^3\cdot 5 \cdot 7} \biggr]
-+ \xi^6 \biggl[\frac{378}{2^3\cdot 3^5\cdot 5^2\cdot 7} \biggr]
++ \xi^6 \biggl[\frac{1}{2^2\cdot 3^2\cdot 5^2} \biggr]
 </math>
    </td>

Joel2: /* Displacement Function for Isothermal LAWE */

2026-10-13T23:06:50Z

Displacement Function for Isothermal LAWE

← Older revision		Revision as of 18:06, 13 October 2026
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	<td align="left">		<td align="left">
	<math>		<math>
	- \frac{1}{3} + \xi^2\biggl[- \frac{5}{~~2\cdot 3^2 \cdot 5} + \frac{3}{2\cdot~~ 3^2\cdot 5} \biggr]		- \frac{1}{3} - \xi^2\biggl[\frac{1}{3^2 \cdot 5} \biggr]
	+ \xi^4\biggl[-\frac{7}{~~2^2 \cdot 3^3\cdot 5 \cdot 7} + \frac{3\cdot 7}{2^2\cdot 3^3\cdot 5 \cdot 7} -\frac{2^2\cdot 3}{2^~~2\cdot 3^3 \cdot 5\cdot 7} \biggr]		+ \xi^4\biggl[ \frac{1}{2 \cdot 3^3\cdot 5 \cdot 7} \biggr]
	+ \xi^6 \biggl[\frac{~~2\cdot 5}{2^3\cdot 3^5\cdot 5^2\cdot 7} + \frac{3^2 \cdot 7}{2^3 \cdot 3^5 \cdot 5^2}~~		+ \xi^6 \biggl[\frac{378}{2^3\cdot 3^5\cdot 5^2\cdot 7} \biggr]
	~~+ \frac{5\cdot 61~~ }{2^3 \cdot 3^5 \cdot 5^2 \cdot 7}\biggr]
	</math>		</math>
	</td>		</td>

70.180.56.157: /* Displacement Function for Isothermal LAWE */

2026-10-13T22:58:59Z

Displacement Function for Isothermal LAWE

@@ Line 3,194: / Line 3,194: @@
 + \xi^4\biggl[-\frac{1}{2^2 \cdot 3^3\cdot 5} + \frac{1}{2^2\cdot 3^2\cdot 5}  -\frac{1}{3^2 \cdot 5\cdot 7} \biggr]
 + \xi^6 \biggl[\frac{1}{2^2\cdot 3^5\cdot 5\cdot 7} + \frac{1}{2^3 \cdot 3^3\cdot 5^2}  + \frac{61 }{2^3 \cdot 3^5 \cdot 5 \cdot 7}\biggr]
 </math>
    </td>