scipy.special.ellipkinc

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

1

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

2

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

3

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i