Difference between revisions of "Elliptic K"
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The Elliptic $K$ function is also known as the complete Elliptic integral of the first kind. If $m=k^2$ we define the complete elliptic integral of the first kind, $K$ to be | The Elliptic $K$ function is also known as the complete Elliptic integral of the first kind. If $m=k^2$ we define the complete elliptic integral of the first kind, $K$ to be | ||
$$K(k)=K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-k^2\sin^2 \theta}} d\theta.$$ | $$K(k)=K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-k^2\sin^2 \theta}} d\theta.$$ | ||
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| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Elliptickplot.png|Graph of $K$. | ||
| + | File:Complexelliptickplot.png|Domain coloring of $K$. | ||
| + | </gallery> | ||
| + | </div> | ||
=See Also= | =See Also= | ||
Revision as of 16:48, 25 May 2016
The Elliptic $K$ function is also known as the complete Elliptic integral of the first kind. If $m=k^2$ we define the complete elliptic integral of the first kind, $K$ to be $$K(k)=K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-k^2\sin^2 \theta}} d\theta.$$
Graph of $K$.
Domain coloring of $K$.
See Also
Elliptic E
Incomplete Elliptic K
