SciPy

scipy.special.ellipe

scipy.special.ellipe(m) = <ufunc 'ellipe'>

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 : ndarray

Value of the elliptic integral.

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

Notes

Wrapper for the Cephes [R392] 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.

References

[R392](1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html