Module « scipy.special »

Fonction ellipe - module scipy.special

Signature de la fonction ellipe

Description

ellipe.__doc__

ellipe(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

ellipe(m)

Complete elliptic integral of the second kind

This function is defined as

.. math:: 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.

Notes
-----
Wrapper for the Cephes [1]_ routine `ellpe`.

For `m > 0` the computation uses the approximation,

.. math:: E(m) \approx P(1-m) - (1-m) \log(1-m) Q(1-m),

where :math:`P` and :math:`Q` are tenth-order polynomials.  For
`m < 0`, the relation

.. math:: E(m) = E(m/(m - 1)) \sqrt(1-m)

is used.

The parameterization in terms of :math:`m` follows that of section
17.2 in [2]_. Other parameterizations in terms of the
complementary parameter :math:`1 - m`, modular angle
:math:`\sin^2(\alpha) = m`, or modulus :math:`k^2 = m` are also
used, so be careful that you choose the correct parameter.

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

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