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.

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

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.