Editing Apps/ImamuraHadleyCollaboration (section)

===Singular Sturm-Liouville Problem===

[[Apps/Blaes85SlimLimit#Singular_Sturm-Liouville_Problem|As we have discussed in a separate chapter]], there is a class of eigenvalue problems in the mathematical physics literature that is of the "Singular Sturm-Liouville" type.  These problems are governed by a one-dimensional, 2<sup>nd</sup>-order ODE of the form,
<div align="center" id="SingularSturmLiouville">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\lambda \Upsilon(x) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-(1-x)^{-\alpha}(1+x)^{-\beta} \cdot \frac{d}{dx} \biggl[ (1-x)^{\alpha+1}(1+x)^{\beta+1} \cdot \frac{d\Upsilon(x)}{dx} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(x^2-1)\frac{d^2\Upsilon(x)}{dx^2} + [\alpha - \beta + (\alpha+\beta+2)x]\frac{d\Upsilon(x)}{dx} 
\, ,
</math>
  </td>
</tr>
</table>
</div>
where the pair of exponent values, <math>~(\alpha, \beta) </math>, is set by the specific physical problem, while the eigenfunction, <math>~\Upsilon(x)</math>, and associated eigenfrequency, <math>~\lambda</math>, are to be determined.  For any choice of the pair of exponents, there is an infinite number <math>~(j = 0 \rightarrow \infty)</math> of ''analytically known'' eigenvectors that satisfy this governing ODE; they are referred to as ''Jacobi Polynomials.''  Specifically, the j<sup>th</sup> eigenfunction is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Upsilon_j(x) = J_j^{\alpha,\beta}(x)</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~(1-x)^{-\alpha}(1+x)^{-\beta} \biggl\{
\frac{(-1)^j}{2^j j!} \cdot \frac{d^j}{dx^j}\biggl[ (1-x)^{j+\alpha}(1+x)^{j+\beta} \biggr] 
\biggr\} \, ;</math>
  </td>
</tr>
</table>
</div>
and the corresponding j<sup>th</sup> eigenvalue is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lambda_j^{\alpha,\beta}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~j(j+\alpha+\beta + 1) \, .</math>
  </td>
</tr>
</table>
</div>

Table 3 presents the first three eigenfunctions (j = 0, 1, 2), along with each corresponding eigenfrequency; the figure displayed in the right-most column has been extracted directly from &sect;22 (p. 773) of [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun's (1964)] ''Handbook of Mathematical Functions'' (tenth printing, December 1972, with corrections) and shows the behavior of the lowest five Jacobi polynomials (j = 1, 2, 3, 4, 5, as labeled) over the interval <math>~-1 \le x \le +1</math> and for the specific case of <math>~(\alpha,\beta) = (1.5, - 0.5)</math>.


<div align="center" id="Table3">
<table align="center" border="1" cellpadding="5">
<tr>
  <th align="center" colspan="4"><font size="+1">Table 3:  Example Eigenvector Solutions to the Singular Sturm-Liouville Problem</font></th>
</tr>
<tr>
  <td align="center"><math>~j</math></td>
  <td align="center"><math>~J_j^{\alpha,\beta}(x)</math></td>
  <td align="center"><math>~\lambda_j^{\alpha,\beta}</math></td>
  <td align="center" rowspan="5">[[File:JacobiPolynomialsAS.png|200px|Jacobi Polynomials]]</td>
</tr>
<tr>
  <td align="center">
<math>~0</math>
  </td>
  <td align="center">
<math>~1</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~1</math>
  </td>
  <td align="center">
<math>~\tfrac{1}{2}(\alpha+\beta+2)x + \tfrac{1}{2}(\alpha-\beta)</math>
  </td>
  <td align="center">
<math>~(\alpha+\beta+2)</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>~2</math>
  </td>
  <td align="center">
<math>~
\tfrac{1}{8}(12+7\alpha + \alpha^2 + 7\beta+\beta^2+ 2\alpha\beta ) x^2
+ \tfrac{1}{4}(3\alpha + \alpha^2 - 3\beta-\beta^2) x
+ \tfrac{1}{8}(-4 - \alpha + \alpha^2-\beta + \beta^2 - 2\alpha\beta) 
</math>
  </td>
  <td align="center">
<math>~2(\alpha+\beta+3)</math>
  </td>
</tr>
<tr>
  <td align="left" colspan="3">See also, eqs. (35)-(37) of [http://mathworld.wolfram.com/JacobiPolynomial.html Wolfram MathWorld]; and &sect;22 (p. 773) of  [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun's (1964)] ''Handbook of Mathematical Functions,'' from which the illustration on the right has been extracted. 
</tr>
</table>
</div>


Here we highlight the qualitative similarities between the behavior of the set of Jacobi polynomials [figure extracted from [http://people.math.sfu.ca/~cbm/aands/page_773.htm Abramowitz and Stegun (1964)] and reprinted here in Table 3] and the set of eigenfunctions that describe normal modes of oscillation in homogeneous, self-gravitating spheres [figure extracted from [https://ia600302.us.archive.org/12/items/ThePulsationTheoryOfVariableStars/Rosseland-ThePulsationTheoryOfVariableStars.pdf Rosseland (1964)] and reprinted here in Table 2].  As we shall presently see, Jacobi polynomials play an important role in illuminating the structure of normal modes of oscillation in Papaloizou-Pringle tori.