maxframe.tensor.special.ellipeinc#

maxframe.tensor.special.ellipeinc(phi, m, out=None)[source]#

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:

  • phi (array_like) – amplitude of the elliptic integral.

  • m (array_like) – parameter of the elliptic integral.

  • out (ndarray, optional) – Optional output array for the function values

Returns:

E – Value of the elliptic integral.

Return type:

scalar or ndarray

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