# 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
marray_like

Defines the parameter of the elliptic integral.

Returns
Endarray

Value of the elliptic integral.

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 [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.

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.

Examples

This function is used in finding the circumference of an ellipse with semi-major axis a and semi-minor axis b.

>>> from scipy import special

>>> a = 3.5
>>> b = 2.1
>>> e_sq = 1.0 - b**2/a**2  # eccentricity squared


Then the circumference is found using the following:

>>> C = 4*a*special.ellipe(e_sq)  # circumference formula
>>> C
17.868899204378693


When a and b are the same (meaning eccentricity is 0), this reduces to the circumference of a circle.

>>> 4*a*special.ellipe(0.0)  # formula for ellipse with a = b
21.991148575128552
>>> 2*np.pi*a  # formula for circle of radius a
21.991148575128552


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