Appendix/Mathematics/Hypergeometric - Revision history

Jet53man: /* See Also */

2022-12-04T18:58:06Z

Jet53man: /* Second Try */

2022-12-01T01:27:01Z

Second Try

← Older revision		Revision as of 21:27, 30 November 2022
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	</tr>		</tr>
	</table>		</table>
			The coefficient of the first-derivative term also may be rewritten ad,

	==Uniform Density==		==Uniform Density==

Jet53man: /* Second Try */

2022-12-01T01:07:52Z

Second Try

Show changes

Jet53man: /* Second Try */

2022-11-30T19:39:36Z

Second Try

← Older revision		Revision as of 15:39, 30 November 2022
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	<math>		<math>
	+ \frac{1}{2}		+ \frac{1}{2}
	\biggl[ (2B-6)F^{4/B} - ( 3B + 19) F^{2/B} \biggl( G+z \biggr)^{2/B} + (B+9)\biggl( G+z \biggr)^{4/B}\biggr]		\biggl[ (2B+6)F^{4/B} - ( 3B + 19) F^{2/B} \biggl( G+z \biggr)^{2/B} + (B+9)\biggl( G+z \biggr)^{4/B}\biggr]
	\cdot \frac{df}{dz}		\cdot \frac{df}{dz}
	</math>		</math>
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	\cdot \frac{d^2f}{dz^2}		\cdot \frac{d^2f}{dz^2}
	+ \frac{1}{2}		+ \frac{1}{2}
	\biggl[ -2 - 25 \biggl( 1+z \biggr) + 11\biggl( 1+z \biggr)^{2}\biggr]		\biggl[ 10 - 25 \biggl( 1+z \biggr) + 11\biggl( 1+z \biggr)^{2}\biggr]
	\cdot \frac{df}{dz}		\cdot \frac{df}{dz}
	+ \frac{1}{2}\biggl[		+ \frac{1}{2}\biggl[
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	\cdot \frac{d^2f}{dz^2}		\cdot \frac{d^2f}{dz^2}
	+ \frac{1}{2}		+ \frac{1}{2}
	\biggl[ -2 - 25 - 25z + 11\biggl( 1+ 2z + z^2 \biggr)\biggr]		\biggl[ 10 - 25 - 25z + 11\biggl( 1+ 2z + z^2 \biggr)\biggr]
	\cdot \frac{df}{dz}		\cdot \frac{df}{dz}
	+ \frac{1}{2}\biggl[		+ \frac{1}{2}\biggl[
Line 4,691:		Line 4,691:
	z(z-1) ( 1 + z )\cdot \frac{d^2f}{dz^2}		z(z-1) ( 1 + z )\cdot \frac{d^2f}{dz^2}
	+ \frac{1}{2}		+ \frac{1}{2}
	\biggl[ -16 - 3z + 11 z^2 \biggr]		\biggl[ -4 - 3z + 11 z^2 \biggr]
	\cdot \frac{df}{dz}		\cdot \frac{df}{dz}
	+ \frac{1}{2}\biggl[		+ \frac{1}{2}\biggl[
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	<table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">		<table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">
	<div align="center"><b>LAWE from "First Try"<b></div>		<div align="center"><b>Compare with LAWE from "First Try"<b></div>
	<table border=0 cellpadding=2 align="center">		<table border=0 cellpadding=2 align="center">

Jet53man: /* Second Try */

2022-11-30T00:11:57Z

Second Try

← Older revision		Revision as of 20:11, 29 November 2022
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	<td align="left">		<td align="left">
	<math>		<math>
	BF\biggl( \frac{G + z}{F} \biggr)^{(B-1) / B}		BF\biggl( \frac{G + z}{F} \biggr)^{(B-1) / B} = 2(1+z)^{1 / 2}
	\, ;		\, ;
	</math>		</math>
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	<math>		<math>
	BF\biggl( \frac{G + z}{F} \biggr)^{(B-1) / B}\frac{df}{dz}		BF\biggl( \frac{G + z}{F} \biggr)^{(B-1) / B}\frac{df}{dz}
			=
			2(1+z)^{1 / 2} \cdot \frac{df}{dz}
	\, ;		\, ;
	</math>		</math>
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	B( G + z ) \cdot \frac{d^2f}{dz^2}		B( G + z ) \cdot \frac{d^2f}{dz^2}
	\biggr]		\biggr]
			=
			2(1+z)^{0}\biggl[\frac{df}{dz} + 2(1+z) \frac{d^2f}{dz^2} \biggr]
	\, .		\, .
	</math>		</math>

Jet53man: /* Second Try */

2022-11-30T00:03:57Z

Second Try

@@ Line 4,238: / Line 4,238: @@
 This should reduce to the "First Try" example by setting:  &nbsp;<math>(B, F, G) = (2, 1, 1)</math>.  Let's see &hellip;
-<table border=0 cellpadding=2 align="center">
+<tr>
 <tr>
    <td align="right">
-<math>0</math>
+<math>
    </td>
    <td align="center">
@@ Line 4,250: / Line 4,281: @@
    <td align="left">
 <math>
-\frac{1}{2}\biggl[ 2 - 3  \biggl( 1+z \biggr) + \biggl( 1+z \biggr)^{2}\biggr]
+BF^{2/B}\biggl( G+z \biggr)^{1 - 2/B}\biggl[
-( 1 + z )
+(B-1) \cdot \frac{df}{dz}
-\cdot \frac{d^2f}{dz^2}
-+ \frac{1}{2}
+B( G + z )  \cdot \frac{d^2f}{dz^2}
-\biggl[ -2 - 25 \biggl( 1+z \biggr) + 11\biggl( 1+z \biggr)^{2}\biggr]
+\biggr]
-\cdot \frac{df}{dz}
+\, .
 </math>
    </td>
@@ Line 4,270: / Line 4,298: @@
 <tr>
    <td align="right">
-<math>0</math>
+<math>z = x^2 - 1</math>
    </td>
    <td align="center">
@@ Line 4,277: / Line 4,349: @@
    <td align="left">
 <math>
--2z(1 - z^2) \cdot \frac{d^2 f}{dz^2}
+\biggl[ 2 \cdot \frac{df}{dz}  + 4(1+z) \cdot \frac{d^2 f}{dz^2}\biggr] \, .
 </math>
    </td>

Jet53man: /* Second Try */

2022-11-29T23:55:26Z

Second Try

@@ Line 4,236: / Line 4,236: @@
 </tr>
 </table>
 Hence, the ''parabolic'' LAWE takes the form,
@@ Line 4,577: / Line 4,632: @@
 <table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">
+<div align="center"><b>LAWE from "First Try"<b></div>
 <table border=0 cellpadding=2 align="center">

Jet53man: /* Parabolic Density Distribution */

2022-11-29T00:22:48Z

Parabolic Density Distribution

← Older revision		Revision as of 20:22, 28 November 2022
Line 4,508:		Line 4,508:

	Now, this should reduce to the "First Try" example by setting:  <math>(B, F, G) = (2, 1, 1)</math>. Let's see …		Now, this should reduce to the "First Try" example by setting:  <math>(B, F, G) = (2, 1, 1)</math>. Let's see …

			<table border=0 cellpadding=2 align="center">

			<tr>
			<td align="right">
			<math>0</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\frac{1}{2}\biggl[ 2 - 3 \biggl( 1+z \biggr) + \biggl( 1+z \biggr)^{2}\biggr]
			2( 1 + z )
			\cdot \frac{d^2f}{dz^2}
			+ \frac{1}{2}
			\biggl[ -2 - 25 \biggl( 1+z \biggr) + 11\biggl( 1+z \biggr)^{2}\biggr]
			\cdot \frac{df}{dz}
			+ \frac{1}{2}\biggl[
			\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( 1+z \biggr)
			\biggr] \cdot f
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\biggl[ 2 - 3 -3z + \biggl( 1+2z +z^2\biggr)\biggr]
			( 1 + z )
			\cdot \frac{d^2f}{dz^2}
			+ \frac{1}{2}
			\biggl[ -2 - 25 - 25z + 11\biggl( 1+ 2z + z^2 \biggr)\biggr]
			\cdot \frac{df}{dz}
			+ \frac{1}{2}\biggl[
			\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( 1+z \biggr)
			\biggr] \cdot f
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			z(z-1) ( 1 + z )\cdot \frac{d^2f}{dz^2}
			+ \frac{1}{2}
			\biggl[ -16 - 3z + 11 z^2 \biggr]
			\cdot \frac{df}{dz}
			+ \frac{1}{2}\biggl[
			\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( 1+z \biggr)
			\biggr] \cdot f \, .
			</math>
			</td>
			</tr>
			</table>

			<table border="1" width="80%" cellpadding="10" align="center"><tr><td align="left">

			<table border=0 cellpadding=2 align="center">

			<tr>
			<td align="right">
			<math>0</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-2z(1 - z^2) \cdot \frac{d^2 f}{dz^2}
			+ \biggl[
			-4 - 3z + 11z^2
			\biggr] \cdot \frac{df}{dz}
			+ \biggl[
			\biggl( \frac{5}{2} \cdot \mathfrak{F} + 3\alpha\biggr) + 3\alpha z
			\biggr] \cdot f
			</math>
			</td>
			</tr>
			</table>

			</td></tr></table>

	==Uniform Density==		==Uniform Density==

Jet53man: /* Parabolic Density Distribution */

2022-11-28T23:59:37Z

Parabolic Density Distribution

← Older revision		Revision as of 19:59, 28 November 2022
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	B( G + z )		B( G + z )
	\cdot \frac{d^2f}{dz^2}		\cdot \frac{d^2f}{dz^2}
			+ B^{-1}( G+z )^{-1 + 2/B}\biggl[
			F^{2/B}\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( G+z \biggr)^{2/B}
			\biggr] \cdot f
	</math>		</math>
	</td>		</td>
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	<math>		<math>
	+ \frac{1}{2}		+ \frac{1}{2}
	\biggl\{ \biggl[ 2F^{4/B} - 3 F^{2/B} \biggl( G+z \biggr)^{2/B} + ~~2\frac{1}{2}~~\biggl( G+z \biggr)^{4/B}\biggr]		\biggl\{ \biggl[ 2F^{4/B} - 3 F^{2/B} \biggl( G+z \biggr)^{2/B} + \biggl( G+z \biggr)^{4/B}\biggr]
	(B-1)		(B-1)
	+ \biggl[		+ \biggl[
Line 4,473:		Line 4,476:
	</td>		</td>
	<td align="center">		<td align="center">
	~~ ~~		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
			\frac{1}{2}\biggl[ 2F^{4/B} - 3 F^{2/B} \biggl( G+z \biggr)^{2/B} + \biggl( G+z \biggr)^{4/B}\biggr]
			B( G + z )
			\cdot \frac{d^2f}{dz^2}
	+ B^{-1}( G+z )^{-1 + 2/B}\biggl[		+ B^{-1}( G+z )^{-1 + 2/B}\biggl[
	F^{2/B}\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( G+z \biggr)^{2/B}		F^{2/B}\biggl( \frac{5}{2} \biggr) \mathfrak{F} + 3\alpha \biggl( G+z \biggr)^{2/B}
	\biggr] \cdot f		\biggr] \cdot f
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			+ \frac{1}{2}
			\biggl[ (2B-6)F^{4/B} - ( 3B + 19) F^{2/B} \biggl( G+z \biggr)^{2/B} + (B+9)\biggl( G+z \biggr)^{4/B}\biggr]
			\cdot \frac{df}{dz}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

			Now, this should reduce to the "First Try" example by setting:  <math>(B, F, G) = (2, 1, 1)</math>. Let's see …

	==Uniform Density==		==Uniform Density==

Jet53man: /* Parabolic Density Distribution */

2022-11-28T19:54:37Z

Parabolic Density Distribution

@@ Line 4,412: / Line 4,412: @@
 + \biggl[
 F^{4/B} - 11 F^{2/B}\biggl( G+z \biggr)^{2/B}  + 5 \biggl( G+z \biggr)^{4/B}  \biggr]
 \biggr\} \cdot \frac{df}{dz}
 </math>

← Older revision		Revision as of 14:58, 4 December 2022
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	<ul>		<ul>
			<li>
			Our "Ramblings" chapter titled, [[SSC/Structure/OtherAnalyticModels\|Other Analytically Definable, Spherical Equilibrium Structures]]
			</li>
	<li>		<li>
	In an article titled, "Radial Oscillations of a Stellar Model," {{ Prasad49full }} investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,		In an article titled, "Radial Oscillations of a Stellar Model," {{ Prasad49full }} investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,