maxframe.tensor.special.ellipkm1#

maxframe.tensor.special.ellipkm1(p, out=None)[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.
out (ndarray, optional) – Optional output array for the function values

Returns:

K – Value of the elliptic integral.

Return type:

scalar or ndarray

See also

ellipk: Complete elliptic integral of the first kind
ellipkinc: Incomplete elliptic integral of the first kind
ellipe: Complete elliptic integral of the second kind
ellipeinc: Incomplete elliptic integral of the second kind
elliprf: Completely-symmetric elliptic integral of the first kind.

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