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Module « scipy.special »

Fonction hyp2f1 - module scipy.special

Signature de la fonction hyp2f1

def hyp2f1(*args, **kwargs) 

Description

help(scipy.special.hyp2f1)

hyp2f1(x1, x2, x3, x4, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])


    hyp2f1(a, b, c, z, out=None)

    Gauss hypergeometric function 2F1(a, b; c; z)

    Parameters
    ----------
    a, b, c : array_like
        Arguments, should be real-valued.
    z : array_like
        Argument, real or complex.
    out : ndarray, optional
        Optional output array for the function values

    Returns
    -------
    hyp2f1 : scalar or ndarray
        The values of the gaussian hypergeometric function.

    See Also
    --------
    hyp0f1 : confluent hypergeometric limit function.
    hyp1f1 : Kummer's (confluent hypergeometric) function.

    Notes
    -----
    This function is defined for :math:`|z| < 1` as

    .. math::

       \mathrm{hyp2f1}(a, b, c, z) = \sum_{n=0}^\infty
       \frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!},

    and defined on the rest of the complex z-plane by analytic
    continuation [1]_.
    Here :math:`(\cdot)_n` is the Pochhammer symbol; see `poch`. When
    :math:`n` is an integer the result is a polynomial of degree :math:`n`.

    The implementation for complex values of ``z`` is described in [2]_,
    except for ``z`` in the region defined by

    .. math::

         0.9 <= \left|z\right| < 1.1,
         \left|1 - z\right| >= 0.9,
         \mathrm{real}(z) >= 0

    in which the implementation follows [4]_.

    References
    ----------
    .. [1] NIST Digital Library of Mathematical Functions
           https://dlmf.nist.gov/15.2
    .. [2] S. Zhang and J.M. Jin, "Computation of Special Functions", Wiley 1996
    .. [3] Cephes Mathematical Functions Library,
           http://www.netlib.org/cephes/
    .. [4] J.L. Lopez and N.M. Temme, "New series expansions of the Gauss
           hypergeometric function", Adv Comput Math 39, 349-365 (2013).
           https://doi.org/10.1007/s10444-012-9283-y

    Examples
    --------
    >>> import numpy as np
    >>> import scipy.special as sc

    It has poles when `c` is a negative integer.

    >>> sc.hyp2f1(1, 1, -2, 1)
    inf

    It is a polynomial when `a` or `b` is a negative integer.

    >>> a, b, c = -1, 1, 1.5
    >>> z = np.linspace(0, 1, 5)
    >>> sc.hyp2f1(a, b, c, z)
    array([1.        , 0.83333333, 0.66666667, 0.5       , 0.33333333])
    >>> 1 + a * b * z / c
    array([1.        , 0.83333333, 0.66666667, 0.5       , 0.33333333])

    It is symmetric in `a` and `b`.

    >>> a = np.linspace(0, 1, 5)
    >>> b = np.linspace(0, 1, 5)
    >>> sc.hyp2f1(a, b, 1, 0.5)
    array([1.        , 1.03997334, 1.1803406 , 1.47074441, 2.        ])
    >>> sc.hyp2f1(b, a, 1, 0.5)
    array([1.        , 1.03997334, 1.1803406 , 1.47074441, 2.        ])

    It contains many other functions as special cases.

    >>> z = 0.5
    >>> sc.hyp2f1(1, 1, 2, z)
    1.3862943611198901
    >>> -np.log(1 - z) / z
    1.3862943611198906

    >>> sc.hyp2f1(0.5, 1, 1.5, z**2)
    1.098612288668109
    >>> np.log((1 + z) / (1 - z)) / (2 * z)
    1.0986122886681098

    >>> sc.hyp2f1(0.5, 1, 1.5, -z**2)
    0.9272952180016117
    >>> np.arctan(z) / z
    0.9272952180016122
    

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