Editing Appendix/Ramblings/Bordeaux (section)

====Keeping Higher Order in Wong's First Component====


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

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  E(k_0) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 ~+~\frac{1}{2^5} ~k_0^4 
~+~\frac{1}{2^5} ~ k_0^6 
~+~\frac{231}{2^{13}} ~ k_0^8 
+ \mathcal{O}(k_0^{10}) \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot  K(k_0) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{1}{2} k_0^2
+ \frac{11}{2^5} ~k_0^4 
+ \frac{17}{2^6} ~ k_0^6 
+ \frac{1787}{2^{13}} ~k^8
+ \mathcal{O}(k_0^{10}) 
\, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot  E(k_0) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 
~-~\frac{1}{2^6} ~ k_0^6
~-~\frac{77}{2^{13}} ~ k_0^8
+ \mathcal{O}(k_0^{10}) \, .
</math>
  </td>
</tr>
</table>

<table border="1" width="80%" cellpadding="5" align="center">
<tr><td align="center">'''Add One Additional Term'''</td></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  E(k) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 + \biggl( \frac{1}{2} \biggr)^2k^2 
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 
+ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
+ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} 
~-~ \cdots
 \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
~-~ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} 
\biggr\}
~+~
\biggl\{
\biggl( \frac{1}{2} \biggr)^2k^2 
\biggr\}
\times~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~+~
\biggl\{
\biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 
\biggr\}
\times~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
\biggr\}
~+~
\biggl\{
\biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 
\biggr\}\times
\biggl\{1 - \frac{1}{2^2} ~k^2 
\biggr\}
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \biggl(\frac{5}{2^8}\biggr)~k^6
- \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)~k^8
\biggr\}
~+~
\biggl\{
\biggl( \frac{1}{2^2} \biggr)k^2 
- \frac{1}{2^4} ~k^4 
- \frac{3}{2^8}~ k^6 
-\frac{5}{2^{10}} ~k^8
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~
\biggl( \frac{3^2}{2^6}\biggr) k^4 
~-~
\biggl( \frac{3^2}{2^8}\biggr) k^6 
~-~ \biggl(\frac{3^3}{2^{12}}\biggr) ~k^8
~+~
\biggl( \frac{5^2}{2^8}\biggr) k^6 
~-~\biggl(\frac{5^2}{2^{10}}\biggr) ~k^8
~+~\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) ~k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 
~+~\biggl[ \frac{1}{2^2}  
- \frac{1}{2^2} \biggr] ~k^2 
~+~\biggl[ \frac{3^2}{2^6} 
- \frac{3}{2^6} 
- \frac{1}{2^4} \biggr]~k^4 
~+~\biggl[ \frac{5^2}{2^8}  
- \frac{5}{2^8}
- \frac{3}{2^8} 
~-~\frac{3^2}{2^8} \biggr]~ k^6 
+ \biggl[
\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) - \biggl(\frac{5^2}{2^{10}}\biggr) - \biggl(\frac{3^3}{2^{12}}\biggr) - \frac{5}{2^{10}} - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)
\biggr]~k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 ~+~\frac{1}{2^5} ~k^4 
~+~\frac{1}{2^5} ~ k^6 
~+~\frac{231}{2^{13}} ~ k^8 
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot  K(k) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 + \biggl( \frac{1}{2} \biggr)^2k^2 
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 
+ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
+ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times~
\biggl\{
1 + \biggl( \frac{1}{2} \biggr)^2k^2 
+ \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 
+ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 
+ \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 
+ \cdots
+ \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n}
+ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 
+ \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
+ \frac{5^2}{2^8} k^6 
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8 
\biggr\}
\times~
\biggl\{
1 
+ \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
+ \frac{5^2}{2^8} k^6 
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8 
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 
+ \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
+ \frac{5^2}{2^8} k^6 
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8 
\biggr\}
~+~\frac{1}{2^2} k^2 
\biggl\{
1 
+ \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
+ \frac{5^2}{2^8} k^6 
\biggr\}
~+~\frac{3^2}{2^6} k^4 
\biggl\{
1 
+ \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
\biggr\}
~+~
\biggl\{
\frac{5^2}{2^8} k^6 
\biggr\}
\biggl\{
1 
+ \frac{1}{2^2} k^2 
\biggr\}
~+~
\biggl\{
\frac{5^2 \cdot 7^2}{2^{14}} k^8
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 + \frac{1}{2^2} k^2 
+ \frac{3^2}{2^6} k^4 
+ \frac{5^2}{2^8} k^6 
+ \frac{5^2 \cdot 7^2}{2^{14}} k^8 
\biggr\}
~+~ 
\biggl\{
\frac{1}{2^2} k^2
+ \frac{1}{2^4} k^4 
+ \frac{3^2}{2^8} k^6 
+ \frac{5^2}{2^{10}} k^8
\biggr\}
~+~
\biggl\{
\frac{3^2}{2^6} k^4 
~+~\frac{3^2}{2^8} k^6 
~+~\frac{3^4}{2^{12}} k^8
\biggr\}
~+~
\biggl\{
\frac{5^2}{2^8} k^6 
~+~\frac{5^2}{2^{10}} k^8
\biggr\}
~+~
\biggl\{
\frac{5^2 \cdot 7^2}{2^{14}} k^8
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \biggl[ \frac{1}{2^2} 
~+~ \frac{1}{2^2} \biggr] ~k^2
+ \biggl[ \frac{3^2}{2^6} 
+ \frac{1}{2^4} 
~+~\frac{3^2}{2^6} \biggr]~k^4 
+ \biggl[ \frac{5^2}{2^8} 
+ \frac{3^2}{2^8} 
~+~\frac{3^2}{2^8}  
~+~\frac{5^2}{2^8} \biggr]~ k^6 
~+~\biggl[
\frac{5^2 \cdot 7^2}{2^{14}}+ \frac{5^2}{2^{10}} ~+~\frac{3^4}{2^{12}} ~+~\frac{5^2}{2^{10}} ~+~\frac{5^2 \cdot 7^2}{2^{14}}
\biggr]k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{1}{2} k^2
+ \frac{11}{2^5} ~k^4 
+ \frac{17}{2^6} ~ k^6 
+ \frac{1787}{2^{13}} ~k^{8}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>
</table>


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

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[E(k) \cdot  E(k) \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} 
~-~ \cdots
 \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
~-~ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
\times~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 
- \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5}
- \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} 
~-~ \cdots
 \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1}
~-~ \cdots
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \frac{5}{2^8}~ k^6
- \frac{5^2\cdot 7}{2^{14}} ~k^8
\biggr\}
\times
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \frac{5}{2^8}~ k^6
- \frac{5^2\cdot 7}{2^{14}} ~k^8
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \frac{5}{2^8}~ k^6
- \frac{5^2\cdot 7}{2^{14}} ~k^8
\biggr\}
~+~
\biggl\{- \frac{1}{2^2} ~k^2 
\biggr\}
\times
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \frac{5}{2^8}~ k^6
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~
\biggl\{
- \frac{3}{2^6}~ k^4 
\biggr\}
\times
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
\biggr\}
~+~
\biggl\{
- \frac{5}{2^8}~ k^6
\biggr\}
\times
\biggl\{
1 - \frac{1}{2^2} ~k^2 
\biggr\}
~+~
\biggl\{
- \frac{5^2\cdot 7}{2^{14}}k^8
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>



<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{
1 - \frac{1}{2^2} ~k^2 
- \frac{3}{2^6}~ k^4 
- \frac{5}{2^8}~ k^6
- \frac{5^2\cdot 7}{2^{14}} ~k^8
\biggr\}
~+~
\biggl\{
- \frac{1}{2^2} ~k^2  + \frac{1}{2^4} ~k^4 
+ \frac{3}{2^8}~ k^6 
+ \frac{5}{2^{10}}~k^8
\biggr\}
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~
\biggl\{
~-~ \frac{3}{2^6}~ k^4 
~+~ \frac{3}{2^8}~ k^6 
+ \frac{3^2}{2^{12}}~k^8
\biggr\}
~+~
\biggl\{
~-~\frac{5}{2^8}~ k^6
+ \frac{5}{2^{10}}~k^8
\biggr\}
~+~
\biggl\{
~-~ \frac{5^2\cdot 7}{2^{14}}k^8
\biggr\}
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>


<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \biggl[ - \frac{1}{2^2}- \frac{1}{2^2} \biggr]k^2
+ \biggl[ - \frac{3}{2^6}+ \frac{1}{2^4}~-~ \frac{3}{2^6} \biggr] k^4
+ \biggl[ - \frac{5}{2^8}+ \frac{3}{2^8}~+~ \frac{3}{2^8}~-~\frac{5}{2^8} \biggr] k^6
+ \biggl[ - \frac{5^2\cdot 7}{2^{14}} + \frac{5}{2^{10}}+ \frac{3^2}{2^{12}}+ \frac{5}{2^{10}}~-~ \frac{5^2\cdot 7}{2^{14}} \biggr] k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \biggl[ - \frac{2}{2^2} \biggr]k^2
+ \biggl[ - \frac{3}{2^5}+ \frac{2}{2^5} \biggr] k^4
+ \biggl[ - \frac{5}{2^7}+ \frac{3}{2^7} \biggr] k^6
+ \biggl[ - \frac{5^2\cdot 7}{2^{13}} + \frac{5\cdot 2^4}{2^{13}}+ \frac{2 \cdot 3^2}{2^{13}}\biggr] k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 ~-~\frac{77}{2^{13}} ~ k^8
+ \mathcal{O}(k^{10})
</math>
  </td>
</tr>
</table>

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


Next, employing the [[Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k_0^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e(1+e)^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2e(  1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k_0^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4e^2(1+e)^{-2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4e^2(  1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k_0^6</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3e^3(1+e)^{-3}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^3e^3(  1 - 3e + 6e^2 - 10e^3 + 15e^4 - 21e^5 + \cdots ) \, ;</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~k_0^8</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^4e^4(1+e)^{-4}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2^4e^4(1 - 4e + 10 e^2 - 20e^3 + 35e^4 - 56e^5 + \cdots) \, .</math>
  </td>
</tr>
</table>

<span id="Step03">Hence, we have,</span>

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

<tr>
  <td align="right">
<math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (1-e) \biggl[ 1 + \frac{1}{2} k_0^2
+ \frac{11}{2^5} ~k_0^4 
+ \frac{17}{2^6} ~ k_0^6 
+ \frac{1787}{2^{13}} ~k_0^8
+ \mathcal{O}(k_0^{10}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 
~+~\frac{1}{2^5} ~ k_0^6 
~+~ \frac{231}{2^{13}}~k_0^8
+ \mathcal{O}(k_0^{10}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~
~-~\frac{1}{2^6} ~ k_0^6
~-~\frac{77}{2^{13}}~k_0^8
+ \mathcal{O}(k_0^{10}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- (1-e) \biggl[ 1 + e (  1 - e +e^2 -e^3 )
+ \frac{11}{2^3} \cdot ~e^2(  1 - 2e  + 3e^2)
+ \frac{17}{2^6} ~\cdot 2^3 e^3 ( 1 - 3e)
+ \frac{1787}{2^{13}} \cdot 2^4e^4
+ \mathcal{O}(e^{5}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} \cdot~4e^2(  1 - 2e  + 3e^2) 
~+~\frac{1}{2^5} ~ 2^3e^3 (1-3e) 
~+~\frac{231}{2^{13}} \cdot 2^4e^4
+ \mathcal{O}(e^{5}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- (1+e) \biggl[ 1 - e ~(  1 - e +e^2 -e^3) ~
-~ \frac{1}{2^5} \cdot~ 4e^2(  1 - 2e  +3e^2) ~
~-~\frac{1}{2^6} ~ 2^3e^3(1 - 3e)
~-~\frac{77}{2^{13}}~2^4e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{-9}(1-e) \biggl[ 512 (1+  e - e^2 +e^3 -e^4 )
+ 704  \cdot ~(  e^2 - 2e^3  + 3e^4)
+ 1088 ~\cdot ( e^3 - 3e^4)
+ 1787 \cdot e^4
+ \mathcal{O}(e^{5}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (1+e^2) 2^{-9}\biggl[ 1024 
~+~128 \cdot~(  e^2 - 2e^3  + 3e^4) 
~+~256 ~ (e^3-3e^4) 
~+~462 \cdot e^4
+ \mathcal{O}(e^{5}) 
 \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{-9}(1+e) \biggl[ 512 ~(1  -e + e^2 -e^3 + e^4) ~
-~ 64 \cdot~ (  e^2 - 2e^3  +3e^4) ~
~-~64 ~ (e^3 - 3e^4)
~-~77~e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{-9}(1-e) \biggl[
512 + 512e + 192e^2 + e^3(512 - 1408 + 1088) + e^4(704-512 - 3264 + 1787)
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2^{-9}(1+e^2) \biggl[
1024 + 128e^2 + e^3(-256 + 256  ) + e^4(384 -768 + 462)
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{-9}(1+e) \biggl[
512 - 512e + e^2(512 - 64  ) + e^3(-512 +128 -64  ) + e^4(512 - 192 - 192 - 77)
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- 2^{-9}(1-e) \biggl[
512 + 512e + 192e^2 + 192e^3 - 1285 e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2^{-9}(1+e^2) \biggl[
1024 + 128e^2 + 78e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- 2^{-9}(1+e) \biggl[
512 - 512e + 448e^2 - 448 e^3 + 51e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2^{-9}\biggl[
(- 192+ 128 - 448)e^2  
+ (- 192 + 448) e^3 
+ (1285 + 78- 51)e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ 2^{-9} e \biggl[
1024e + (192- 448)e^2  
+ (192+ 448) e^3  
+ \mathcal{O}(e^{4}) 
\biggr]
+ 2^{-9} e^2 \biggl[
1024 + 128e^2 
+ \mathcal{O}(e^{3}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2^{-9} \biggl[ 
(- 192+ 128 - 448)e^2  + 2048 e^2 
+ (1285 + 78- 51)e^4
+ (192+ 448) e^4  
+ 128e^4 
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2^{-9} \biggl[ 
1536e^2  
+ 2080e^4
+ \mathcal{O}(e^{5}) 
\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3e^2  
+ \biggl[ \frac{5\cdot 13}{2^4}\biggr] e^4
+ \mathcal{O}(e^{5}) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, .
</math>
  </td>
</tr>
</table>

Hence,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
[1 + \mathcal{O}(e^{2})]\cdot (1 - e^2)^{1 / 2}
\biggl\{
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, ,
</math>
  </td>
</tr>
</table>
or, more precisely,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2}
\biggl\{
\frac{\boldsymbol{K}(k) }{ r_1 }  \biggr\}\Delta_0  \, .
</math>
  </td>
</tr>
</table>