SciPy

scipy.special.ellipkinc

scipy.special.ellipkinc(phi, m) = <ufunc 'ellipkinc'>

Incomplete elliptic integral of the first kind

This function is defined as

\[K(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{-1/2} dt\]

This function is also called F(phi, m).

Parameters:

phi : array_like

amplitude of the elliptic integral

m : array_like

parameter of the elliptic integral

Returns:

K : ndarray

Value of the elliptic integral

See also

ellipkm1
Complete elliptic integral of the first kind, near m = 1
ellipk
Complete 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 [R431] routine ellik. The computation is carried out using the arithmetic-geometric mean algorithm.

The parameterization in terms of \(m\) follows that of section 17.2 in [R432]. Other parameterizations in terms of the complementary parameter \(1 - m\), modular angle \(\sin^2(\alpha) = m\), or modulus \(k^2 = m\) are also used, so be careful that you choose the correct parameter.

References

[R431](1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html
[R432](1, 2) Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

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