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