scipy.special.ellipeinc

scipy.special.ellipeinc(phi, m, out=None) = <ufunc 'ellipeinc'>

Incomplete elliptic integral of the second kind

This function is defined as

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

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

Escalar 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

ellipkinc

Incomplete elliptic integral of the first kind

ellipe

Complete elliptic integral of the second kind

elliprd

Symmetric elliptic integral of the second kind.

elliprf

Completely-symmetric elliptic integral of the first kind.

elliprg

Completely-symmetric elliptic integral of the second kind.

Notes

Wrapper for the Cephes [1] routine ellie.

Computation uses arithmetic-geometric means 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 E incomplete integral can be related to combinations of Carlson’s symmetric integrals R_D, R_F, and R_G in multiple ways [3]. For example, with \(c = \csc^2\phi\),

\[E(\phi, m) = R_F(c-1, c-k^2, c) - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .\]

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