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

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

Notes

Wrapper for the Cephes [R348] 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

[R348]

(1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html