Editing Appendix/Ramblings/PowerSeriesExpressions (section)

===Isothermal Lane-Emden Function===

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

<tr>
  <td align="right">
<math>~\frac{1}{\xi^2}\frac{d}{d\xi}\biggl( \xi^2 \frac{d\psi}{d\xi}\biggr)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-\psi} \, .</math>
  </td>
</tr>
</table>
</div>
-->

Here we seek a power-series expression for the isothermal, Lane-Emden function &#8212; expanded about the coordinate center &#8212; 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">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-w} \, . </math>
  </td>
</tr>
</table>
</div>

A general power-series should be of the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
w_0 + ar + br^2 + cr^3 + dr^4 + er^5 + fr^6 + gr^7 + hr^8 +\cdots
</math>
  </td>
</tr>
</table>
</div>

Derivatives:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{dw}{dr}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7 +\cdots \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{d^2w}{dr^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6 +\cdots \, .
</math>
  </td>
</tr>
</table>
</div>

Put together, then, the left-hand-side of the isothermal Lane-Emden equation becomes:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{d^2w}{dr^2} +\frac{2}{r} \frac{d w}{dr} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2b + 2\cdot 3cr + 2^2\cdot 3dr^2 + 2^2\cdot 5er^3 + 2\cdot 3 \cdot 5fr^4 + 2\cdot 3 \cdot 7gr^5 + 2^3\cdot 7hr^6
+ \frac{2}{r}\biggl[ a + 2br + 3cr^2 + 4dr^3 + 5er^4 + 6fr^5 + 7gr^6 + 8hr^7  \biggr] + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2a}{r} + r^0(6b) + r^1(2^2\cdot 3c) + r^2(2^2\cdot 3d + 2^3d) + r^3(2^2\cdot 5e + 2\cdot 5e)
+ r^4(2\cdot 3\cdot 5 f + 2^2\cdot 3f) + r^5(2\cdot 3\cdot 7 g+ 2\cdot 7g) + r^6(2^3 \cdot 7 h + 2^4 h) + \cdots
</math>
  </td>
</tr>
</table>
</div>

Drawing on the [[#Exponential|above power-series expression for an exponential function]], and adopting the convention that <math>~w_0 = 0</math>, the right-hand-side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~e^{-w}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
e^{0}\cdot e^{-ar} \cdot e^{-br^2} \cdot e^{-cr^3} \cdot e^{-dr^4} \cdot e^{-er^5} \cdot e^{-fr^6} \cdot e^{-gr^7} \cdot e^{-hr^8} \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 -ar + \frac{a^2r^2}{2!} - \frac{a^3r^3}{3!} + \frac{a^4r^4}{4!} - \frac{a^5r^5}{5!} + \frac{a^6r^6}{6!} + \cdots \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl[ 1 -br^2 + \frac{b^2r^4}{2!} - \frac{b^3r^6}{3!} + \cdots \biggr] \times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2!} + \cdots \biggr]
\times \biggl[1 - dr^4\biggr] \times \biggl[1 - er^5\biggr]\times \biggl[1 - fr^6\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr]
\times \biggl[ 1 -cr^3 + \frac{c^2r^6}{2}  -br^2 + bcr^5 + \frac{b^2r^4}{2}  - \frac{b^3r^6}{6} \biggr] 
\times \biggl[1 - dr^4 - er^5 - fr^6\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl\{
\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} \biggr]
- dr^4 \biggl[ 1 -ar + \frac{a^2r^2}{2} \biggr] - er^5 \biggl[ 1 -ar \biggr] - fr^6
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl[ 1  -br^2 -cr^3  + \frac{b^2r^4}{2}  + bcr^5  + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + \frac{a^4r^4}{24} - \frac{a^5r^5}{5\cdot 24} + \frac{a^6r^6}{30\cdot 24} 
- dr^4  + adr^5 - \frac{a^2d r^6}{2}   - er^5   +  aer^6  - fr^6
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times \biggl[ 1  -br^2 -cr^3  + \frac{b^2r^4}{2}  + bcr^5  + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl[
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) + r^5\biggl(ad - e-\frac{a^5}{5\cdot 24}\biggr) 
+ r^6 \biggl(\frac{a^6}{30\cdot 24} - \frac{a^2d}{2}   +  ae  - f \biggr)
\biggr]
\times \biggl[ 1  -br^2 -cr^3  + \frac{b^2r^4}{2}  + bcr^5  + r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) + r^5\biggl(ad - e-\frac{a^5}{5\cdot 24}\biggr) 
+ r^6 \biggl(\frac{a^6}{30\cdot 24} - \frac{a^2d}{2}   +  ae  - f \biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~-br^2\biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} + r^4\biggl(\frac{a^4}{24} - d\biggr) \biggr]
-cr^3  \biggl[ 1 -ar + \frac{a^2r^2}{2} - \frac{a^3r^3}{6} \biggr]
+ \frac{b^2r^4}{2}\biggl[ 1 -ar + \frac{a^2r^2}{2} \biggr]
+  bcr^5\biggl[1 -ar \biggr]
+ r^6\biggl(\frac{c^2}{2}- \frac{b^3}{6}\biggr)  
</math>
  </td>
</tr>
</table>
</div>

Expressions for the various coefficients can now  be determined by equating terms on the LHS and RHS that have like powers of <math>~r</math>.  Beginning with the highest order terms, we initially find,
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">Term</td>
  <td align="center">LHS</td>
  <td align="center">RHS</td>
  <td align="center">Implication</td>
</tr>

<tr>
  <td align="right">
<math>~r^{-1}:</math>
  </td>
  <td align="center">
<math>~2a</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~a=0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{0}:</math>
  </td>
  <td align="center">
<math>~6b</math>
  </td>
  <td align="center">
<math>~1</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~b = + \frac{1}{6}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{1}:</math>
  </td>
  <td align="center">
<math>~2^2\cdot 3c</math>
  </td>
  <td align="center">
<math>~-a</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~c = -\frac{a}{2^2\cdot 3} =0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{2}:</math>
  </td>
  <td align="center">
<math>~(2^2\cdot 3d + 2^3d)</math>
  </td>
  <td align="center">
<math>~\frac{a^2}{2} - b</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~d = \frac{1}{20}\biggl( \frac{a^2}{2} - b \biggr) = - \frac{1}{120}</math>
  </td>
</tr>
</table>
</div>

With this initial set of coefficient values in hand, we can rewrite (and significantly simplify) our approximate expression for the RHS, namely,

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

<tr>
  <td align="right">
<math>~e^{-w}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
1  -d r^4 -e r^5 -f r^6 
-br^2 ( 1 -d r^4 ) + \frac{b^2r^4}{2} - \frac{b^3r^6}{6} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 -br^2+ r^4 \biggl(\frac{b^2}{2}  -d \biggr) -e r^5  
+r^6\biggl( bd - \frac{b^3}{6} -f \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>

Continuing, then, with equating terms with like powers on both sides of the equation, we find,
<div align="center">
<table border="1" cellpadding="5" align="center">
<tr>
  <td align="center">Term</td>
  <td align="center">LHS</td>
  <td align="center">RHS</td>
  <td align="center">Implication</td>
</tr>

<tr>
  <td align="right">
<math>~r^{3}:</math>
  </td>
  <td align="center">
<math>~30e</math>
  </td>
  <td align="center">
<math>~0</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~e=0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{4}:</math>
  </td>
  <td align="center">
<math>~(2\cdot 3\cdot 5 f + 2^2\cdot 3f)</math>
  </td>
  <td align="center">
<math>~\biggl(\frac{b^2}{2}  -d \biggr) </math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~f = \frac{1}{2\cdot 3\cdot 7}\biggl(\frac{1}{2^3\cdot 3^2}+\frac{1}{2^3\cdot 3 \cdot 5}\biggr) = \frac{1}{2\cdot 3^3\cdot 5 \cdot 7}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{5}:</math>
  </td>
  <td align="center">
<math>~(2\cdot 3\cdot 7 g+ 2\cdot 7g)</math>
  </td>
  <td align="center">
<math>~-e</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~g = 0</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~r^{6}:</math>
  </td>
  <td align="center">
<math>~(2^3 \cdot 7 h + 2^4 h)</math>
  </td>
  <td align="center">
<math>~\biggl( bd - \frac{b^3}{6} -f \biggr)</math>
  </td>
  <td align="left">
<math>~\Rightarrow ~~~
h = -\frac{1}{2^3\cdot 3^2}\biggl(  \frac{1}{2^4\cdot 3^2 \cdot 5} + \frac{1}{2^4\cdot 3^4} + \frac{1}{2\cdot 3^3\cdot 5\cdot 7}\biggr) 
= -\frac{61}{2^{6} \cdot  3^6\cdot 5\cdot 7}
</math>
  </td>
</tr>
</table>
</div>



Result:
<div align="center" id="IsothermalLaneEmden">
<table border="1" width="80%" cellpadding="8" align="center">
<tr><th align="center">For Spherically Symmetric Configurations</th></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~w(r) 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{r^2}{6} - \frac{r^4}{120} + \frac{r^6}{1890} - \frac{61 r^8}{1,632,960} + \cdots \, .</math>
  </td>
</tr>
</table>
</td></tr></table>
</div>


See also:
* Equation (377) from &sect;22 in Chapter IV of [[Appendix/References#C67|C67]].


NOTE:  &nbsp;For cylindrically symmetric, rather than spherically symmetric, configurations, an analytic expression for the function, <math>~w(r)</math>, is presented as equation (56) in a paper by [http://adsabs.harvard.edu/abs/1964ApJ...140.1056O J. P. Ostriker (1964, ApJ, 140, 1056)] titled, ''The Equilibrium of Polytropic and Isothermal Cylinders''.