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 - 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