# 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  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 . 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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