# 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 K : ndarray Value of the elliptic integral

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 [1] 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 [2]. 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

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