scipy.special.ellipe#

scipy.special.ellipe(m, out=None) = <wrapped_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,

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

Array API Standard Support

ellipe has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. ellipe does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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