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Module « scipy.special »

Fonction ellipeinc - module scipy.special

Signature de la fonction ellipeinc

def ellipeinc(*args, **kwargs) 

Description

help(scipy.special.ellipeinc)

ellipeinc(x1, x2, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])


    ellipeinc(phi, m, out=None)

    Incomplete elliptic integral of the second kind

    This function is defined as

    .. math:: E(\phi, m) = \int_0^{\phi} [1 - m \sin(t)^2]^{1/2} dt

    Parameters
    ----------
    phi : array_like
        amplitude of the elliptic integral.
    m : array_like
        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
    ellipe : Complete elliptic integral of the second kind
    elliprd : Symmetric elliptic integral of the second kind.
    elliprf : Completely-symmetric elliptic integral of the first kind.
    elliprg : Completely-symmetric elliptic integral of the second kind.

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

    Computation uses arithmetic-geometric means algorithm.

    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 incomplete integral can be related to combinations
    of Carlson's symmetric integrals R_D, R_F, and R_G in multiple
    ways [3]_. For example, with :math:`c = \csc^2\phi`,

    .. math::
      E(\phi, m) = R_F(c-1, c-k^2, c)
        - \frac{1}{3} k^2 R_D(c-1, c-k^2, c) .

    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
    

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