Module « scipy.special »

Fonction gegenbauer - module scipy.special

Signature de la fonction gegenbauer

def gegenbauer(n, alpha, monic=False) 

Description

gegenbauer.__doc__

Gegenbauer (ultraspherical) polynomial.

    Defined to be the solution of

    .. math::
        (1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)}
          - (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)}
          + n(n + 2\alpha)C_n^{(\alpha)} = 0

    for :math:`\alpha > -1/2`; :math:`C_n^{(\alpha)}` is a polynomial
    of degree :math:`n`.

    Parameters
    ----------
    n : int
        Degree of the polynomial.
    alpha : float
        Parameter, must be greater than -0.5.
    monic : bool, optional
        If `True`, scale the leading coefficient to be 1. Default is
        `False`.

    Returns
    -------
    C : orthopoly1d
        Gegenbauer polynomial.

    Notes
    -----
    The polynomials :math:`C_n^{(\alpha)}` are orthogonal over
    :math:`[-1,1]` with weight function :math:`(1 - x^2)^{(\alpha -
    1/2)}`.

    Examples
    --------
    >>> from scipy import special
    >>> import matplotlib.pyplot as plt

    We can initialize a variable ``p`` as a Gegenbauer polynomial using the
    `gegenbauer` function and evaluate at a point ``x = 1``.

    >>> p = special.gegenbauer(3, 0.5, monic=False)
    >>> p
    poly1d([ 2.5,  0. , -1.5,  0. ])
    >>> p(1)
    1.0

    To evaluate ``p`` at various points ``x`` in the interval ``(-3, 3)``,
    simply pass an array ``x`` to ``p`` as follows:

    >>> x = np.linspace(-3, 3, 400)
    >>> y = p(x)

    We can then visualize ``x, y`` using `matplotlib.pyplot`.

    >>> fig, ax = plt.subplots()
    >>> ax.plot(x, y)
    >>> ax.set_title("Gegenbauer (ultraspherical) polynomial of degree 3")
    >>> ax.set_xlabel("x")
    >>> ax.set_ylabel("G_3(x)")
    >>> plt.show()