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.
See also
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.