SciPy

scipy.special.ellipk

scipy.special.ellipk(m)[source]

Complete elliptic integral of the first kind.

This function is defined as

\[K(m) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt\]
Parameters:

m : array_like

The parameter of the elliptic integral.

Returns:

K : array_like

Value of the elliptic integral.

See also

ellipkm1
Complete elliptic integral of the first kind around m = 1
ellipkinc
Incomplete elliptic integral of the first kind
ellipe
Complete elliptic integral of the second kind
ellipeinc
Incomplete elliptic integral of the second kind

Notes

For more precision around point m = 1, use ellipkm1, which this function calls.

The parameterization in terms of \(m\) follows that of section 17.2 in [R458]. Other parameterizations in terms of the complementary parameter \(1 - m\), modular angle \(\sin^2(\alpha) = m\), or modulus \(k^2 = m\) are also used, so be careful that you choose the correct parameter.

References

[R458](1, 2) Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

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