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====Phase 0C==== Referencing our [[2DStructure/ToroidalGreenFunction#Series_Expansions|separate listing of complete elliptic integral series expansions]] drawn from [http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], we have, <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\frac{2}{\pi} \biggl[K(k) - 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 + \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </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} ~-~ \cdots \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} ~-~ \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ \biggl( \frac{1}{2^2} \biggr)k^2 + \biggl( \frac{3^2}{2^6}\biggr)2 k^4 + \biggl( \frac{5^2}{2^8}\biggr) k^6 \biggr\} ~+~ \biggl\{\frac{1}{2^2} ~k^2 + \frac{3}{2^6}~ k^4 + \biggl(\frac{5}{2^8}\biggr)~k^6 \biggr\} + \mathcal{O}(k^8) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ \biggl( \frac{1}{2^2} +\frac{1}{2^2} \biggr)k^2 + \biggl( \frac{3^2}{2^6} + \frac{3}{2^6}\biggr) k^4 + \biggl( \frac{5^2}{2^8} + \frac{5}{2^8} \biggr) k^6 \biggr\} + \mathcal{O}(k^8) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ \biggl( \frac{1}{2} \biggr)k^2 + \biggl( \frac{3}{2^4} \biggr) k^4 + \biggl( \frac{3 \cdot 5}{2^7} \biggr) k^6 \biggr\} + \mathcal{O}(k^8) </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[K(k) - E(k) \biggr]^2</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ \biggl( \frac{1}{2} \biggr)k^2 + \biggl( \frac{3}{2^4} \biggr) k^4 + \biggl( \frac{3 \cdot 5}{2^7} \biggr) k^6 \biggr\} \times \biggl\{ \biggl( \frac{1}{2} \biggr)k^2 + \biggl( \frac{3}{2^4} \biggr) k^4 + \biggl( \frac{3 \cdot 5}{2^7} \biggr) k^6 \biggr\} + \mathcal{O}(k^{10}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ \biggl( \frac{1}{2^2} \biggr)k^4 + \biggl( \frac{3}{2^5} \biggr) k^6 + \biggl( \frac{3 \cdot 5}{2^8} \biggr) k^8 \biggr\} ~+~ \biggl\{ \biggl( \frac{3}{2^5} \biggr)k^6 + \biggl( \frac{3^2}{2^8} \biggr) k^8 \biggr\} ~+~\biggl\{ \biggl( \frac{3 \cdot 5}{2^8} \biggr) k^8 \biggr\} + \mathcal{O}(k^{10}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl( \frac{1}{2^2} \biggr)k^4 + \biggl( \frac{3}{2^4} \biggr) k^6 + \biggl( \frac{3 \cdot 13}{2^8} \biggr) 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 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 + \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </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} ~-~ \cdots \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} ~-~ \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </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} \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 \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </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 \biggr\} ~+~ \biggl\{ \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 \biggr\} ~+~ \biggl\{ \biggl( \frac{1}{2^2} \biggr)k^2 - \frac{1}{2^4} ~k^4 - \frac{3}{2^8}~ k^6 \biggr\} ~+~ \biggl( \frac{3^2}{2^6}\biggr) k^4 ~-~ \biggl( \frac{3^2}{2^8}\biggr) k^6 ~+~ \biggl( \frac{5^2}{2^8}\biggr) k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 ~+~\frac{1}{2^5} ~k^4 ~+~\frac{1}{2^5} ~ k^6 + \mathcal{O}(k^{8}) </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 + \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </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 + \cdots + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} + \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </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 \biggr\} \times~ \biggl\{ 1 + \frac{1}{2^2} k^2 + \frac{3^2}{2^6} k^4 + \frac{5^2}{2^8} k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 \biggr\} ~+~\frac{1}{2^2} k^2 \biggl\{ 1 + \frac{1}{2^2} k^2 + \frac{3^2}{2^6} k^4 \biggr\} ~+~\frac{3^2}{2^6} k^4 \biggl\{ 1 + \frac{1}{2^2} k^2 \biggr\} ~+~ \biggl\{ \frac{5^2}{2^8} k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 + \frac{1}{2^2} k^2 + \frac{3^2}{2^6} k^4 + \frac{5^2}{2^8} k^6 ~+~ \frac{1}{2^2} k^2 + \frac{1}{2^4} k^4 + \frac{3^2}{2^8} k^6 ~+~\frac{3^2}{2^6} k^4 ~+~\frac{3^2}{2^8} k^6 ~+~ \frac{5^2}{2^8} k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 + \mathcal{O}(k^{8}) </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} ~-~ \cdots \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} ~-~ \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> </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} ~-~ \cdots \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} ~-~ \cdots \biggr\} </math> </td> </tr> <tr> <td align="right"> </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 \biggr\} \times \biggl\{ 1 - \frac{1}{2^2} ~k^2 - \frac{3}{2^6}~ k^4 - \frac{5}{2^8}~ k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 \biggr\} ~+~ \biggl\{- \frac{1}{2^2} ~k^2 \biggr\} \times \biggl\{ 1 - \frac{1}{2^2} ~k^2 - \frac{3}{2^6}~ k^4 \biggr\} ~+~ \biggl\{ - \frac{3}{2^6}~ k^4 \biggr\} \times \biggl\{ 1 - \frac{1}{2^2} ~k^2 \biggr\} ~+~ \biggl\{ - \frac{5}{2^8}~ k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2^2} ~k^2 - \frac{3}{2^6}~ k^4 - \frac{5}{2^8}~ k^6 - \frac{1}{2^2} ~k^2 + \frac{1}{2^4} ~k^4 + \frac{3}{2^8}~ k^6 ~-~ \frac{3}{2^6}~ k^4 ~+~ \frac{3}{2^8}~ k^6 ~-~\frac{5}{2^8}~ k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ 1 - \frac{1}{2} ~k^2 + \biggl[ \frac{1}{2^4} - \frac{3}{2^6} ~-~ \frac{3}{2^6} \biggr]~ k^4 + \biggl[ \frac{3}{2^8} ~+~ \frac{3}{2^8} - \frac{5}{2^8}~-~\frac{5}{2^8} \biggr] ~ k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </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 + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> <math>~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[E(k) \cdot E(k) - K(k) \cdot K(k)\biggr]</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ \biggl\{ 1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 \biggr\} ~-~ \biggl\{ 1 + \frac{1}{2} k^2+ \frac{11}{2^5} ~k^4 + \frac{17}{2^6} ~ k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> <tr> <td align="right"> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~ ~-~\biggl\{ k^2 ~+~ \frac{3}{2^3} ~ k^4 ~+~\frac{3^2}{2^5} ~ k^6 \biggr\} + \mathcal{O}(k^{8}) </math> </td> </tr> </table> <table border="1" align="center" cellpadding="8"> <tr> <th align="center">Summary</th> </tr> <tr> <td align="left"> <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>~ 1 ~+~\frac{1}{2^5} ~k^4 ~+~\frac{1}{2^5} ~ k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <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>~ 1 + \frac{1}{2} k^2 + \frac{11}{2^5} ~k^4 + \frac{17}{2^6} ~ k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> <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>~ 1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 + \mathcal{O}(k^{8}) </math> </td> </tr> </table> </td> </tr> </table>
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