Editing Apps/DysonPotential (section)

===Alternate "Small" Argument of Elliptic Integrals===

<!--
Let's switch from <math>~4K(\mu)/(R_1+R)</math> to <math>~2K(k)/R_1</math>, where, <math>~k \equiv [1 - (R/R_1)^2]^{1 / 2}</math>, and recognize that,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k' = \sqrt{1-k^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~e^{-\eta} = \frac{R}{R_1} \, .</math>
  </td>
</tr>
</table>

-->
Defining the "small parameter,"
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\sqrt{1-\mu^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[1 - \biggl( \frac{e^\eta - 1}{e^\eta + 1}\biggr)^2  \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(e^\eta + 1)^2 - (e^\eta - 1)^2 }{(e^\eta + 1)^2}  \biggr]^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\frac{(e^{2\eta} + 2e^\eta+ 1)- (e^{2\eta} - 2e^\eta + 1) }{(e^\eta + 1)^2}  \biggr]^{1 / 2}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{R}{R_1}\biggr)^{1 / 2}
\biggl[1 + \frac{R}{R_1}  \biggr]^{-1} \, .
</math>
  </td>
</tr>
</table>
At the surface of the torus, where <math>~R=a</math>, we therefore have,

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

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2}
\biggl[1 + \frac{a}{c}\cdot  \frac{c}{R_1}  \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2}
\biggl[1 + \frac{a}{c}\cdot  \frac{c}{R_1}  \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2}
\biggl[  4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]^{1 / 4}
\biggl\{ 1 + \frac{a}{c}\cdot  \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]^{-1 / 2} \biggr\} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2}
\biggl[  1  - \biggl(\frac{a}{c}\biggr)\cos\chi  + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4}
\biggl\{ 1 + \frac{a}{2c}\cdot  \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi  + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr)
+ \frac{1}{4}\ln\biggl[  1  - \biggl(\frac{a}{c}\biggr)\cos\chi  \biggr]
+ \ln\biggl[ 1 + \frac{a}{2c} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr)
+ \frac{a}{2c}\biggl[ 1
- \frac{1}{2} \cos\chi \biggr]
</math>
  </td>
</tr>
</table>

Alternatively, if &#8212; [[#Higher_Order|as above]] &#8212; we adopt the shorthand notation, <math>~\gamma \equiv R_1/(2c)</math>, we can write,

<table border="1" cellpadding="8" width="85%" align="center">
<tr><td align="left">
<div align="center">'''Summary'''</div>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2  (1-\cos^2\chi)
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1  ~+~ 6 \cos^2\chi   ~-~ 5 \cos^4\chi   \biggr]
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{1}{\gamma}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>
</table>

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

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

<tr>
  <td align="right">
<math>~k'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2}
\biggl[1 + \biggl(\frac{a}{c}\biggr)  \frac{1}{2\gamma}  \biggr]^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2}
\biggl[1 + \biggl(\frac{a}{c}\biggr)  \frac{1}{\gamma} + \biggl(\frac{a}{c}\biggr)^2  \frac{1}{4\gamma^2}  \biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^{1 / 2} 
\biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2  \frac{1}{\gamma}  \biggr]^{-1 / 2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~(k')^{2m}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^{m } 
\biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2  \frac{1}{\gamma}  \biggr]^{-m } 
=
\biggl( \frac{2a}{c}\biggr)^{m } \Gamma^{-m}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} 
\biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2  \frac{1}{\gamma}  \biggr]^{1 / 2} 
=
\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \Gamma^{1 / 2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~\ln \frac{4}{k'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma \, ,
</math>
  </td>
</tr>
</table>

where,

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

<tr>
  <td align="right">
<math>~\Gamma</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2  \frac{1}{\gamma} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi  + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2  (1-\cos^2\chi)
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1  ~+~ 6 \cos^2\chi   ~-~ 5 \cos^4\chi   \biggr]
~+~ \biggl(\frac{a}{c}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2  \biggl\{
1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] 
+  \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi 
~-~  3\cos\chi \biggr]
~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3
~-~ 30 \cos^2\chi
~+~ 35 \cos^4\chi \biggr]
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 
+ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi 
+ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 3\cos^2\chi - 1 ) 
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
1 + \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5  ~+~ 18 \cos^2\chi   ~-~ 5 \cos^4\chi )
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, .
</math>
  </td>
</tr>
</table>

Now, if we adopt the shorthand notation,
<div align="center">
<math>~g \equiv 
\biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5  ~+~ 18 \cos^2\chi   ~-~ 5 \cos^4\chi )
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ,
</math>
</div>
we also have,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\ln\Gamma = \ln (1 + g)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
g - \frac{1}{2}g^2 + \frac{1}{3}g^3 - \frac{1}{4}g^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5  ~+~ 18 \cos^2\chi   ~-~ 5 \cos^4\chi )
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi)
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5  ~+~ 18 \cos^2\chi   ~-~ 5 \cos^4\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
- \frac{1}{2} \biggl\{
\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2  
+ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi )
+~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~  \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi)  (1~-~ \frac{1}{2} \cos\chi )  
~+~  \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi)  (3-\cos^2\chi) 
~+~  \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(1~-~ \frac{1}{2} \cos\chi )  
\biggr\}
</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )  
+ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) 
~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi)
~-~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi )
~-~  \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi)  (2~-~ \cos\chi )  
~+~\frac{1}{2^3\cdot 3}\biggl(\frac{a}{c}\biggr)^3 (2~-~ \cos\chi )^2 (2~-~\cos\chi )
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~-~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi )
~-~  \frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi)  (3-\cos^2\chi) 
~-~  \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi )  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
~+~ \frac{1}{2^4\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (2~-~ \cos\chi ) (3-\cos^2\chi)(2~-~ \cos\chi )
+\frac{1}{2^5\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi)  (2~-~ \cos\chi )^2 
+~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5  ~+~ 18 \cos^2\chi   ~-~ 5 \cos^4\chi )
- \frac{1}{2^6}  \biggl(\frac{a}{c}\biggr)^4(2~-~  \cos\chi )^4  
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>
</table>

And,

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

<tr>
  <td align="right">
<math>~~(k')^{2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr) \Gamma^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr) (1+g)^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr) \biggl\{1 
~-~ g 
~+~ g^2 
~-~ g^3 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>

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

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

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

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

<tr>
  <td align="right">
<math>~~(k')^{4}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^2 \Gamma^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^2 (1+g)^{-2}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 
~-~ 2g 
~+~ 3g^2 
\biggr\}
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4\biggl( \frac{a}{c}\biggr)^2 
-~\biggl(\frac{a}{c}\biggr)^3 (8~-~ 4\cos\chi )  
~+~ \biggl(\frac{a}{c}\biggr)^4 ( 9 - 12\cos\chi + 4\cos^2\chi )
~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) 
</math>
  </td>
</tr>
</table>