scipy.special.ellipkm1#
- scipy.special.ellipkm1(p, out=None) = <ufunc 'ellipkm1'>#
Complete elliptic integral of the first kind around m = 1
This function is defined as
\[K(p) = \int_0^{\pi/2} [1 - m \sin(t)^2]^{-1/2} dt\]where m = 1 - p.
- Parameters:
- parray_like
Defines the parameter of the elliptic integral as m = 1 - p.
- outndarray, optional
Optional output array for the function values
- Returns:
- Kscalar or ndarray
Value of the elliptic integral.
See also
Notes
Wrapper for the Cephes [1] routine ellpk.
For p <= 1, computation uses the approximation,
\[K(p) \approx P(p) - \log(p) Q(p),\]where \(P\) and \(Q\) are tenth-order polynomials. The argument p is used internally rather than m so that the logarithmic singularity at m = 1 will be shifted to the origin; this preserves maximum accuracy. For p > 1, the identity
\[K(p) = K(1/p)/\sqrt(p)\]is used.
References
[1]Cephes Mathematical Functions Library, http://www.netlib.org/cephes/