scipy.special.ellipk

scipy.special.ellipk(m, out=None) = <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.

outndarray, optional

Optional output array for the function values

Returns:

Kscalar or ndarray

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

elliprf

Completely-symmetric elliptic integral of the first 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.

The Legendre K integral is related to Carlson’s symmetric R_F function by [2]:

\[K(m) = R_F(0, 1-k^2, 1) .\]

References

[1]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[2]

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i