SciPy

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 [R349] 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

[R349](1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html

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