scipy.special.ellipe#

scipy.special.ellipe(m, out=None) = <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.
outndarray, optional: Optional output array for the function values

Returns:

Escalar or 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
elliprd: Symmetric elliptic integral of the second kind.
elliprg: Completely-symmetric 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.

The Legendre E integral is related to Carlson’s symmetric R_D or R_G functions in multiple ways [3]. For example,

\[E(m) = 2 R_G(0, 1-k^2, 1) .\]

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.

[3]

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.28 of 2020-09-15. See Sec. 19.25(i) https://dlmf.nist.gov/19.25#i

Examples

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

>>> import numpy as np
>>> 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