scipy.special.ellipkm1¶
- scipy.special.ellipkm1(p) = <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: p : array_like
Defines the parameter of the elliptic integral as m = 1 - p.
Returns: K : ndarray
Value of the elliptic integral.
See also
Notes
Wrapper for the Cephes [R372] 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
[R372] (1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html