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
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The elliptic $K$ function (also known as the complete elliptic integral of the first kind) is defined by
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$$K(k)=K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-k^2\sin^2 \theta}} d\theta.$$
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$$K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-m\sin^2 \theta}} \mathrm{d}\theta.$$
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The incomplete elliptic integral of the first kind is
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$$K(\phi |k) = K(\phi |m) = \displaystyle\int_0^{\phi} \dfrac{1}{\sqrt{1-k^2\sin^2 \theta}} d\theta.$$
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<div align="center">
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<gallery>
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File:Elliptickplot.png|Graph of $K$.
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File:Complexelliptickplot.png|Domain coloring of $K$.
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</gallery>
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</div>
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=Properties=
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[[K(m)=(pi/2)2F1(1/2,1/2;1;m)]]<br />
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=See Also=
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[[Elliptic E]] <br />
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[[Incomplete Elliptic K]]
  
 
=References=
 
=References=
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[http://web.mst.edu/~lmhall/SPFNS/spfns.pdf "Special Functions" by Leon Hall]
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* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=findme|next=findme}}: $17.3.1$
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[[Category:SpecialFunction]]

Latest revision as of 04:48, 21 December 2017

The elliptic $K$ function (also known as the complete elliptic integral of the first kind) is defined by $$K(m)=\displaystyle\int_0^{\frac{\pi}{2}} \dfrac{1}{\sqrt{1-m\sin^2 \theta}} \mathrm{d}\theta.$$

  • Graph of $K$.

  • Domain coloring of $K$.

Properties

K(m)=(pi/2)2F1(1/2,1/2;1;m)

See Also

Elliptic E
Incomplete Elliptic K

References

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