Template:Math/EQ EllipticIntegral02 - Revision history

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link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals $\frac{2E(k)}{\pi}$ $=$ $1 - \frac{1}{2^2} ~k^2 - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^..."$
2024-06-29T21:55:56Z

Created page with "<table border="0" cellpadding="5" align="center"> <tr> <td align="right"> 25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals </td> <td align="right"> <math>\frac{2E(k)}{\pi}</math> </td> <td align="center"> <math>=</math> </td> <td align="left"> <math> 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^..."

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[[Image:LSUkey.png|25px|link=Appendix/SpecialFunctions#Complete_Elliptic_Integrals]]
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<math>\frac{2E(k)}{\pi}</math>
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<math>=</math>
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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
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[http://www.mathtable.com/gr/ Gradshteyn & Ryzhik (1965)], §8.114.1
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