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Module « scipy.special »

Fonction jv - module scipy.special

Signature de la fonction jv

def jv(*args, **kwargs) 

Description

help(scipy.special.jv)

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


    jv(v, z, out=None)

    Bessel function of the first 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 values

    Returns
    -------
    J : scalar or ndarray
        Value of the Bessel function, :math:`J_v(z)`.

    See Also
    --------
    jve : :math:`J_v` with leading exponential behavior stripped off.
    spherical_jn : spherical Bessel functions.
    j0 : faster version of this function for order 0.
    j1 : faster version of this function for order 1.

    Notes
    -----
    For positive `v` values, the computation is carried out using the AMOS
    [1]_ `zbesj` routine, which exploits the connection to the modified
    Bessel function :math:`I_v`,

    .. math::
        J_v(z) = \exp(v\pi\imath/2) I_v(-\imath z)\qquad (\Im z > 0)

        J_v(z) = \exp(-v\pi\imath/2) I_v(\imath z)\qquad (\Im z < 0)

    For negative `v` values the formula,

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

    is used, where :math:`Y_v(z)` is the Bessel function of the second
    kind, computed using the AMOS routine `zbesy`.  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)`.

    Not to be confused with the spherical Bessel functions (see `spherical_jn`).

    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 jv
    >>> jv(0, 1.)
    0.7651976865579666

    Evaluate the function at one point for different orders.

    >>> jv(0, 1.), jv(1, 1.), jv(1.5, 1.)
    (0.7651976865579666, 0.44005058574493355, 0.24029783912342725)

    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:

    >>> jv([0, 1, 1.5], 1.)
    array([0.76519769, 0.44005059, 0.24029784])

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

    >>> import numpy as np
    >>> points = np.array([-2., 0., 3.])
    >>> jv(0, points)
    array([ 0.22389078,  1.        , -0.26005195])

    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)

    >>> jv(orders, points)
    array([[ 0.22389078,  1.        , -0.26005195],
           [-0.57672481,  0.        ,  0.33905896]])

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

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

    

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