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Module « scipy.special »

Fonction ellipe - module scipy.special

Signature de la fonction ellipe

def ellipe(*args, **kwargs) 

Description

help(scipy.special.ellipe)

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


    ellipe(m, out=None)

    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.
    out : ndarray, optional
        Optional output array for the function values

    Returns
    -------
    E : scalar 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,

    .. 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.

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

    .. math:: 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
    

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