Elliptic K

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

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  Graph of $K$.

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  Domain coloring of $K$.

Properties

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

See Also

Elliptic E
Incomplete Elliptic K

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $17.3.1$