Difference between revisions of "Elliptic E"

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If $m=k^2$ we define the complete elliptic integral of the second kind, $E$, to be
 
If $m=k^2$ we define the complete elliptic integral of the second kind, $E$, to be
$$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} d\theta.$$
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$$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} \mathrm{d}\theta.$$
The incomplete elliptic integral of the second kind is
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$$E(\phi|k)=E(\phi|m)=\displaystyle\int_0^{\phi} \sqrt{1-m\sin^2 \theta}d\theta.$$
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<div align="center">
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<gallery>
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File:Ellipticeplot.png|Graph of $E$.
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File:Complexellipticeplot.png|[[Domain coloring]] of $E$.
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</gallery>
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</div>
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=Properties=
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[[E(m)=(pi/2)2F1(-1/2,1/2;1;m)]]<br />
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=See Also=
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[[Elliptic K]] <br />
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[[Incomplete Elliptic E]]
  
 
=References=
 
=References=
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf "Special Functions" by Leon Hall]
 
[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf "Special Functions" by Leon Hall]
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 +
[[Category:SpecialFunction]]

Latest revision as of 04:54, 21 December 2017

If $m=k^2$ we define the complete elliptic integral of the second kind, $E$, to be $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} \mathrm{d}\theta.$$

Properties

E(m)=(pi/2)2F1(-1/2,1/2;1;m)

See Also

Elliptic K
Incomplete Elliptic E

References

"Special Functions" by Leon Hall