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.
See also
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