mars.tensor.special.ellipkm1#

mars.tensor.special.ellipkm1(x, **kwargs)[source]#

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 – Value of the elliptic integral.
Return type: ndarray

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.