scipy.special.ellipkinc — SciPy v1.15.2 Manual (original) (raw)

scipy.special.ellipkinc(phi, m, out=None) = <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:

phiarray_like

amplitude of the elliptic integral

marray_like

parameter of the elliptic integral

outndarray, optional

Optional output array for the function values

Returns:

Kscalar or 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

elliprf

Completely-symmetric elliptic integral of the first 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.

The Legendre K incomplete integral (or F integral) is related to Carlson’s symmetric R_F function [3]. Setting \(c = \csc^2\phi\),

\[F(\phi, m) = R_F(c-1, c-k^2, c) .\]

References

[2]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.