mars.tensor.special.ellipe#

mars.tensor.special.ellipe(x, **kwargs)[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.
Returns: E – Value of the elliptic integral.
Return type: ndarray

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.