Difference between revisions of "Elliptic E"
From specialfunctionswiki
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The incomplete elliptic integral of the second kind is | The incomplete elliptic integral of the second kind is | ||
$$E(\phi|k)=E(\phi|m)=\displaystyle\int_0^{\phi} \sqrt{1-m\sin^2 \theta}d\theta.$$ | $$E(\phi|k)=E(\phi|m)=\displaystyle\int_0^{\phi} \sqrt{1-m\sin^2 \theta}d\theta.$$ | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Domaincoloringelliptice.png|[[Domain coloring]] of $E(m)$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
=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] | ||
Revision as of 18:02, 25 July 2015
If $m=k^2$ we define the complete elliptic integral of the second kind, $E$, to be $$E(k)=E(m)=\displaystyle\int_0^{\frac{\pi}{2}} \sqrt{1-k^2\sin^2 \theta} d\theta.$$ The incomplete elliptic integral of the second kind is $$E(\phi|k)=E(\phi|m)=\displaystyle\int_0^{\phi} \sqrt{1-m\sin^2 \theta}d\theta.$$
- Domaincoloringelliptice.png
Domain coloring of $E(m)$.
