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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 [R402] 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

[R402]

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