Editing Apps/Ostriker64 (section)

===Series Expansion===
In the context of Ostriker's expression for the potential, we see that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(k')^{-2} \equiv \biggl[ \frac{1}{1-k^2}\biggr]= n^2 + 1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{4R(R+r\cos\phi)}{r^2} + 1
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2R}{r}\biggr)^2 \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr] \, .
</math>
  </td>
</tr>
</table>
Hence, in the vicinity of the ring where <math>~r/R \ll 1</math> and <math>~k'</math> is a "small parameter," we can draw on the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]] and write, 
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(k')^m</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl[ 1 + \frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2\biggr]^{-m / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 -\frac{m}{2} \biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr] 
+ \frac{1}{2}\biggl[ -\frac{m}{2}\biggl( -\frac{m}{2}-1\biggr) \biggr]\biggl[\frac{r}{R}\cos\phi + \biggl(\frac{r}{2R}\biggr)^2 \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{2^m}\biggl( \frac{r}{R}\biggr)^{m} \biggl\{ 1 - \biggl(\frac{m}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{m}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
+ \frac{m}{4}\biggl( \frac{m}{2} + 1\biggr) \biggl[\frac{r}{R}\cos\phi  \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, .
</math>
  </td>
</tr>
</table>
Note, in particular, that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{1}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{R}{r}\biggr) \biggl\{ 1 + \biggl(\frac{1}{2}\biggr) \frac{r}{R}\cos\phi + \biggl(\frac{1}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
- \frac{1}{2^3} \biggl[\frac{r}{R}\cos\phi  \biggr]^2
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{2R}{r} \biggl\{ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr\} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{r}{2R} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] 
\, ;
</math> &nbsp; &nbsp; &nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~(k')^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2^2}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \frac{r}{R}\cos\phi + \frac{1}{2^2}  \biggl(\frac{r}{R}\biggr)^2 (4\cos^2\phi - 1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] 
\, .
</math>
  </td>
</tr>
</table>

Next we recognize that the following series expansion for the ''complete elliptic integral of the first kind'' &#8212; written in terms of the small parameter, <math>~k'</math> &#8212; appears, for example, as eq. (8.113.3) in the Fourth Edition of [http://www.mathtable.com/gr/ Gradshteyn &amp; Ryzhik (1965)]:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K(k)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr){k'}^2
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} \biggr){k'}^4
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl( \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{2}{5\cdot 6} \biggr){k'}^6 + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2
+ \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4
+ \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30}  \biggr){k'}^6 + \cdots
</math>
  </td>
</tr>
</table>
[This series expansion &#8212; up through the term <math>~\mathcal{O}(k'^4)</math> &#8212; appears as equation 24 in Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II].]  Put together, then, Ostriker's expression for the gravitational potential in the ''thin ring'' approximation becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ -\frac{2GM}{\pi r}  k' K(k)  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl[
k' \ln \frac{4}{k^'}  + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^3
+  \cdots
\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] - \frac{1}{4} k'^3 + \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi r}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] 
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{3} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi - \biggl(\frac{3}{2^3}\biggr)  \biggl(\frac{r}{R}\biggr)^2 
+ \frac{3}{4}\biggl( \frac{3}{2} + 1\biggr) \biggl(\frac{r}{R}\cos\phi  \biggr)^2 + \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr) \biggr] 
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{2GM}{\pi R}  \biggl\{ \ln \frac{4}{k^'} \biggl[ k' + \frac{k'^3}{2^2} \biggr] \frac{R}{r}
- \frac{1}{2^5}\biggl( \frac{r}{R}\biggr)^{2} \biggl[ 1 - \biggl(\frac{3}{2}\biggr) \frac{r}{R}\cos\phi + \mathcal{O}\biggl( \frac{r^2}{R^2}\biggr) \biggr] 
+ \mathcal{O}\biggl( \frac{r^5}{R^5}\biggr)
\biggr\}\, ,
</math>
  </td>
</tr>
</table>
where, again, we have recognized that the mass of the thin hoop is, <math>~M = 2\pi\sigma R</math>.  Now,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ k' \biggl[ 1 + \frac{k'^2}{2^2} \biggr] \frac{R}{r}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{1 +  
\frac{1}{2^4} \biggl[ \biggl( \frac{r}{R}\biggr)^{2} - \biggl(\frac{r}{R}\biggr)^3\cos\phi 
+ \mathcal{O}\biggl(\frac{r^4}{R^4} \biggr) \biggr] 
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 )
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl\{\frac{1}{2} +  
\frac{1}{2^5} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr)  
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr] \, ;
</math>
  </td>
</tr>
</table>
and, given that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln[a (1+x)] = \ln a + \ln(1+x)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln a + x - \tfrac{1}{2}x^2 + \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 + \cdots 
</math>
  </td>
</tr>
</table>

we also have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \ln\biggl[ 1 + \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2} \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \frac{1}{3}\biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]^3 + \cdots
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \biggl[\frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 \sin^2\phi \biggr]
- \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\ln\biggl(\frac{8R}{r} \biggr)
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \, .
</math>
  </td>
</tr>
</table>

So our series expansion for Ostriker's "thin ring" potential becomes,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{4}{k^'} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \biggl[ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) + \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]\biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 ( 3 \cos^2\phi  -1 ) + \frac{1}{2^4} \biggl( \frac{r}{R}\biggr)^{2}
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{ \ln \frac{8R}{r} \biggl[ 1 - \frac{r}{2R}\cos\phi  
+ \frac{1}{2^4} \biggl(\frac{r}{R}\biggr)^2 ( 6 \cos^2\phi  - 1) 
+ \mathcal{O}\biggl(\frac{r^3}{R^3} \biggr) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~ 
+ \frac{1}{2} \biggl(\frac{r}{R}\biggr)\cos\phi + \frac{1}{2^3} \biggl(\frac{r}{R}\biggr)^2 (1 - 2\cos^2\phi ) 
- \frac{1}{2^2} \biggl(\frac{r}{R}\biggr)^2\cos^2\phi 
- \frac{1}{2^4}\biggl( \frac{r}{R}\biggr)^{2}  
+ \mathcal{O}\biggl( \frac{r^3}{R^3}\biggr)
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Finally, dropping the explicit mention of all terms <math>~\mathcal{O}(r^3/R^3)</math> and smaller gives the series expansion formulation presented by Ostriker, namely,

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Phi_\mathrm{TR}(r,\phi)\biggr|_\mathrm{JPO}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \ln \frac{8R}{r}  ( 6 \cos^2\phi  - 1) + (1 - 8\cos^2\phi ) 
\biggr] ~+ ~\cdots \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
-\frac{GM}{\pi R}  \biggl\{\ln \frac{8R}{r}
- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
~+~ \frac{r^2}{2^4R^2} \biggl[ \biggl(2\ln\frac{8R}{r} - 3 \biggr) + \biggl( 3\ln\frac{8R}{r} - 4 \biggr)\cos 2\phi
\biggr] ~+ ~\cdots \biggr\} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (25)
  </td>
</tr>
</table>