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Module « scipy.special »

Fonction jve - module scipy.special

Signature de la fonction jve

def jve(*args, **kwargs) 

Description

help(scipy.special.jve)

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


    jve(v, z, out=None)

    Exponentially scaled Bessel function of the first kind of order `v`.

    Defined as::

        jve(v, z) = jv(v, z) * exp(-abs(z.imag))

    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 exponentially scaled Bessel function.

    See Also
    --------
    jv: Unscaled Bessel function of the first kind

    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)`.

    Exponentially scaled Bessel functions are useful for large arguments `z`:
    for these, the unscaled Bessel functions can easily under-or overflow.

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

    Examples
    --------
    Compare the output of `jv` and `jve` for large complex arguments for `z`
    by computing their values for order ``v=1`` at ``z=1000j``. We see that
    `jv` overflows but `jve` returns a finite number:

    >>> import numpy as np
    >>> from scipy.special import jv, jve
    >>> v = 1
    >>> z = 1000j
    >>> jv(v, z), jve(v, z)
    ((inf+infj), (7.721967686709077e-19+0.012610930256928629j))

    For real arguments for `z`, `jve` returns the same as `jv`.

    >>> v, z = 1, 1000
    >>> jv(v, z), jve(v, z)
    (0.004728311907089523, 0.004728311907089523)

    The function can be evaluated for several orders at the same time by
    providing a list or NumPy array for `v`:

    >>> jve([1, 3, 5], 1j)
    array([1.27304208e-17+2.07910415e-01j, -4.99352086e-19-8.15530777e-03j,
           6.11480940e-21+9.98657141e-05j])

    In the same way, the function can be evaluated at several points in one
    call by providing a list or NumPy array for `z`:

    >>> jve(1, np.array([1j, 2j, 3j]))
    array([1.27308412e-17+0.20791042j, 1.31814423e-17+0.21526929j,
           1.20521602e-17+0.19682671j])

    It is also possible to evaluate several orders at several points
    at the same time by providing arrays for `v` and `z` with
    compatible shapes for broadcasting. Compute `jve` for two different orders
    `v` and three points `z` resulting in a 2x3 array.

    >>> v = np.array([[1], [3]])
    >>> z = np.array([1j, 2j, 3j])
    >>> v.shape, z.shape
    ((2, 1), (3,))

    >>> jve(v, z)
    array([[1.27304208e-17+0.20791042j,  1.31810070e-17+0.21526929j,
            1.20517622e-17+0.19682671j],
           [-4.99352086e-19-0.00815531j, -1.76289571e-18-0.02879122j,
            -2.92578784e-18-0.04778332j]])
    

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