# 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

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