# scipy.special.ellipk¶

scipy.special.ellipk(m) = <ufunc 'ellipk'>

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
marray_like

The parameter of the elliptic integral.

Returns
Karray_like

Value of the elliptic integral.

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 [1]. 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

1

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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