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 | ||
| − | $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} d\theta.$$ | + | $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} \mathrm{d}\theta.$$ |
<div align="center"> | <div align="center"> | ||
<gallery> | <gallery> | ||
| − | File: | + | File:Ellipticeplot.png|Graph of $E$. |
| − | File: | + | File:Complexellipticeplot.png|[[Domain coloring]] of $E$. |
</gallery> | </gallery> | ||
</div> | </div> | ||
| + | |||
| + | =Properties= | ||
| + | [[E(m)=(pi/2)2F1(-1/2,1/2;1;m)]]<br /> | ||
=See Also= | =See Also= | ||
| − | [[ | + | [[Elliptic K]] <br /> |
[[Incomplete Elliptic E]] | [[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] | ||
| + | |||
| + | [[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 $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} \mathrm{d}\theta.$$
Graph of $E$.
Domain coloring of $E$.
Properties
See Also
Elliptic K
Incomplete Elliptic E
