maxframe.tensor.special.ellipe#

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

Complete elliptic integral of the second kind

This function is defined as

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

Parameters:

m (array_like) – Defines the 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
ellipeinc: Incomplete elliptic integral of the second kind
elliprd: Symmetric elliptic integral of the second kind.
elliprg: Completely-symmetric elliptic integral of the second kind.

Notes

Wrapper for the Cephes [1] routine ellpe.

For m > 0 the computation uses the approximation,

\[E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),\]

where \(P\) and \(Q\) are tenth-order polynomials. For m < 0, the relation

\[E(m) = E(m/(m - 1)) \sqrt(1-m)\]

is used.

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 integral is related to Carlson’s symmetric R_D or R_G functions in multiple ways [3]. For example,

\[E(m) = 2 R_G(0, 1-k^2, 1) .\]

References