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Module « scipy.special »

Fonction yv - module scipy.special

Signature de la fonction yv

def yv(*args, **kwargs) 

Description

help(scipy.special.yv)

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


    yv(v, z, out=None)

    Bessel function of the second kind of real order and complex argument.

    Parameters
    ----------
    v : array_like
        Order (float).
    z : array_like
        Argument (float or complex).
    out : ndarray, optional
        Optional output array for the function results

    Returns
    -------
    Y : scalar or ndarray
        Value of the Bessel function of the second kind, :math:`Y_v(x)`.

    See Also
    --------
    yve : :math:`Y_v` with leading exponential behavior stripped off.
    y0: faster implementation of this function for order 0
    y1: faster implementation of this function for order 1

    Notes
    -----
    For positive `v` values, the computation is carried out using the
    AMOS [1]_ `zbesy` routine, which exploits the connection to the Hankel
    Bessel functions :math:`H_v^{(1)}` and :math:`H_v^{(2)}`,

    .. math:: Y_v(z) = \frac{1}{2\imath} (H_v^{(1)} - H_v^{(2)}).

    For negative `v` values the formula,

    .. math:: Y_{-v}(z) = Y_v(z) \cos(\pi v) + J_v(z) \sin(\pi v)

    is used, where :math:`J_v(z)` is the Bessel function of the first kind,
    computed using the AMOS routine `zbesj`.  Note that the second term is
    exactly zero for integer `v`; to improve accuracy the second term is
    explicitly omitted for `v` values such that `v = floor(v)`.

    References
    ----------
    .. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
           of a Complex Argument and Nonnegative Order",
           http://netlib.org/amos/

    Examples
    --------
    Evaluate the function of order 0 at one point.

    >>> from scipy.special import yv
    >>> yv(0, 1.)
    0.088256964215677

    Evaluate the function at one point for different orders.

    >>> yv(0, 1.), yv(1, 1.), yv(1.5, 1.)
    (0.088256964215677, -0.7812128213002889, -1.102495575160179)

    The evaluation for different orders can be carried out in one call by
    providing a list or NumPy array as argument for the `v` parameter:

    >>> yv([0, 1, 1.5], 1.)
    array([ 0.08825696, -0.78121282, -1.10249558])

    Evaluate the function at several points for order 0 by providing an
    array for `z`.

    >>> import numpy as np
    >>> points = np.array([0.5, 3., 8.])
    >>> yv(0, points)
    array([-0.44451873,  0.37685001,  0.22352149])

    If `z` is an array, the order parameter `v` must be broadcastable to
    the correct shape if different orders shall be computed in one call.
    To calculate the orders 0 and 1 for an 1D array:

    >>> orders = np.array([[0], [1]])
    >>> orders.shape
    (2, 1)

    >>> yv(orders, points)
    array([[-0.44451873,  0.37685001,  0.22352149],
           [-1.47147239,  0.32467442, -0.15806046]])

    Plot the functions of order 0 to 3 from 0 to 10.

    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots()
    >>> x = np.linspace(0., 10., 1000)
    >>> for i in range(4):
    ...     ax.plot(x, yv(i, x), label=f'$Y_{i!r}$')
    >>> ax.set_ylim(-3, 1)
    >>> ax.legend()
    >>> plt.show()

    

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