Editing Appendix/Mathematics/Hypergeometric (section)

==Uniform Density==

In the case of a uniform-density, incompressible configuration, the [[#Kopal48Expression|Kopal (1948) LAWE]] becomes,
<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{d^2 f}{dx^2} + \frac{(4-\mu)}{x} \cdot \frac{d f}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_0 R^2}{\gamma_\mathrm{g} P_0} \biggr) 
- \frac{\alpha \mu}{x^2} \biggr] f 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[ 4 - \frac{2x^2}{(1-x^2)} \biggr] \frac{d f}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_c R^2}{\gamma_\mathrm{g} P_c} \biggr) \frac{1}{(1-x^2)}
- \biggl(\frac{2\alpha }{1-x^2}\biggr) \biggr] f 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
(1-x^2) \cdot \frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[ 4 - 6x^2 \biggr] \frac{d f}{dx} 
+ \biggl[\biggl(\frac{\omega^2\rho_c R^2}{\gamma_\mathrm{g} P_c} \biggr) 
- 2\alpha \biggr] f \, .
</math>
  </td>
</tr>
</table>
Given that, in the [[SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|equilibrium state]],
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\frac{\rho_c R^2}{P_c}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{6}{4\pi G \rho_c}
</math>
  </td>
</tr>
</table>
we obtain the LAWE derived by {{ Sterne37full }} &#8212; see his equation (1.91) on p. 585 &#8212; namely,
<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>
(1-x^2) \cdot \frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[ 4 - 6x^2 \biggr] \frac{d f}{dx} 
+ \biggl[6\biggl(\frac{\omega^2}{4\pi \gamma_\mathrm{g} G \rho_c} \biggr) 
- 2\alpha \biggr] f 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
(1-x^2) \cdot \frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[ 4 - 6x^2 \biggr] \frac{d f}{dx} 
+ \mathfrak{F} f \, ,
</math>
  </td>
</tr>
</table>
where,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\mathfrak{F}</math>
  </td> 
  <td align="center">
<math>\equiv</math>
  </td> 
  <td align="left">
<math>
\biggl[6\biggl(\frac{\omega^2}{4\pi \gamma_\mathrm{g} G \rho_c} \biggr) 
- 2\alpha \biggr] \, . 
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="8" align="center">
<tr>
  <th align="center">Summary for PowerPoint Slide</th>
</tr>
<tr>
  <td>
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<b>LAWE:</b> &nbsp; &nbsp;<math>0</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
(1-x^2) \cdot \frac{d^2 f}{dx^2} + \frac{1}{x}\biggl[ 4 - 6x^2 \biggr] \frac{d f}{dx} 
+ \mathfrak{F} f \, ,
</math>
  </td>
</tr>
</table>
where,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\mathfrak{F}</math>
  </td> 
  <td align="center">
<math>\equiv</math>
  </td> 
  <td align="left">
<math>
\biggl[\biggl(\frac{\omega^2\rho_c R^2}{\gamma_\mathrm{g} P_c} \biggr) 
- 2\alpha \biggr] \, . 
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>
This also matches, respectively, equations (8) and (9) of {{ Kopal48full }}, aside from what, we presume, is a type-setting error that appears in the numerator of the second term on the RHS of his equation (8): &nbsp;<math>(4 - x^2)</math> appears, whereas it should be <math>(4 - 6x^2)</math>.

In order to see if this differential equation is of the same form as the ''hypergeometric expression'', we'll make the substitution,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>z</math>
  </td> 
  <td align="center">
<math>\equiv</math>
  </td> 
  <td align="left">
<math>
x^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ dz</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
2x dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{df}{dx}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{dz}{dx} \cdot\frac{df}{dz}
=
2x \cdot\frac{df}{dz}
=
2z^{1 / 2} \cdot\frac{df}{dz}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{d^2f}{dx^2}</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
2z^{1 / 2} \cdot \frac{d}{dz}\biggl[ 2z^{1 / 2} \cdot\frac{df}{dz} \biggr]
=
2z^{1 / 2} \biggl[ z^{-1 / 2} \cdot\frac{df}{dz} + 2z^{1 / 2} \cdot\frac{d^2f}{dz^2}\biggr]
=
\biggl[ 2\cdot\frac{df}{dz} + 4z \cdot\frac{d^2f}{dz^2}\biggr] \, ,
</math>
  </td>
</tr>
</table>
in which case the {{ Sterne37 }} LAWE may be rewritten as,
<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>
(1-z) \biggl[ 2\cdot\frac{df}{dz} + 4z \cdot\frac{d^2f}{dz^2}\biggr] + \frac{1}{z^{1 / 2}}\biggl[ 4 - 6z \biggr]2 z^{1 /2} \frac{d f}{dz} 
+ \mathfrak{F} f
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
(1-z) \biggl[ 4z \cdot\frac{d^2f}{dz^2}\biggr] 
+ (1-z) \biggl[ 2\cdot\frac{df}{dz} \biggr] 
+ 2\biggl[ 4 - 6z \biggr] \frac{d f}{dz} 
+ \mathfrak{F} f
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
4z(1-z) \cdot\frac{d^2f}{dz^2} 
+ 2\biggl[ 5 - 7z \biggr] \frac{d f}{dz} 
+ \mathfrak{F} f \, .
</math>
  </td>
</tr>
</table>
This is, indeed, of the hypergeometric form if we set <math>(\alpha, \beta; \gamma ; z)</math>
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\gamma</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{5}{2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(\alpha + \beta +1)</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{7}{2} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\alpha\beta</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
-\frac{\mathfrak{F}}{4} \, .
</math>
  </td>
</tr>
</table>
Combining this last pair of expressions gives,
<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{\mathfrak{F}}{4} - \alpha\biggl[\frac{5}{2}-\alpha \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\alpha^2 - \biggl(\frac{5}{2}\biggr)\alpha -\frac{\mathfrak{F}}{4} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \alpha</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{1}{2}\biggl\{
\frac{5}{2} \pm \biggl[\biggl(\frac{5}{2}\biggr)^2 - \mathfrak{F}  \biggr]^{1 / 2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{5}{4}\biggl\{
1 \pm \biggl[1 - \biggl(\frac{4}{25}\biggr)\mathfrak{F} \biggr]^{1 / 2}
\biggr\} \, ;
</math>
  </td>
</tr>
</table>
and,
<table border=0 cellpadding=2 align="center">

<tr>
  <td align="right">
<math>\beta</math>
  </td> 
  <td align="center">
<math>=</math>
  </td> 
  <td align="left">
<math>
\frac{5}{2} -
\frac{5}{4}\biggl\{
1 \pm \biggl[1 - \biggl(\frac{4}{25}\biggr)\mathfrak{F} \biggr]^{1 / 2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>

===Example &alpha; = -1===
If we set <math>\alpha = -1</math>, then the eigenvector is,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>u_{1} = F\biggl (-1, \frac{7}{2}; \frac{5}{2}; x^2 \biggr)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl[\frac{ \beta}{\gamma} \biggr]x^2
=
1 - \biggl(\frac{ 7}{5} \biggr)x^2 \, ;
</math>
  </td>
</tr>
</table>
and the corresponding eigenfrequency is obtained from the expression,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>-1</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5}{4}\biggl\{
1 \pm \biggl[1 - \biggl(\frac{4}{25}\biggr)\mathfrak{F} \biggr]^{1 / 2}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~-\frac{9}{5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\pm \biggl[1 - \biggl(\frac{4}{25}\biggr)\mathfrak{F} \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\frac{3^4}{5^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl(\frac{4}{25}\biggr)\mathfrak{F} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\mathfrak{F} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{5^2}{4}\biggr)\biggl[ 1 -  \frac{3^4}{5^2} \biggr]
=
\frac{1}{4}\biggl[ 5^2 -  3^4 \biggr]
=
14 \, .
</math>
  </td>
</tr>
</table>
As we have reviewed in a [[SSC/Stability/UniformDensity#Sterne's_Presentation|separate discussion]], this is identical to the eigenvector identified by {{ Sterne37 }} as mode "<math>j=1</math>".

===More Generally===
More generally, in agreement with {{ Sterne37 }}, for any (positive integer) mode number, <math>0 \le j \le \infty</math>, we find,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\alpha_j</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>- j \, ;</math>
&nbsp; &nbsp; &nbsp; 
<math>\beta_j = \frac{5}{2} + j \, ;</math>
&nbsp; &nbsp; &nbsp; 
<math>\gamma = \frac{5}{2} \, ;</math> 
&nbsp; &nbsp; &nbsp; 
<math>\mathfrak{F} = 2j(2j+5) \, .
</math>
  </td>
</tr>
</table>
And, in terms of the hypergeometric function ''series'', the corresponding eigenfunction is,
 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>u_{j} = </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
F\biggl (\alpha_j, \beta_j; \frac{5}{2}; x^2 \biggr) \, .
</math>
  </td>
</tr>
</table>