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Module « scipy.special »

Fonction yve - module scipy.special

Signature de la fonction yve

def yve(*args, **kwargs) 

Description

help(scipy.special.yve)

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


    yve(v, z, out=None)

    Exponentially scaled Bessel function of the second kind of real order.

    Returns the exponentially scaled Bessel function of the second
    kind of real order `v` at complex `z`::

        yve(v, z) = yv(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 results

    Returns
    -------
    Y : scalar or ndarray
        Value of the exponentially scaled Bessel function.

    See Also
    --------
    yv: Unscaled Bessel function of the second kind of real order.

    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)`.

    Exponentially scaled Bessel functions are useful for large `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 `yv` and `yve` for large complex arguments for `z`
    by computing their values for order ``v=1`` at ``z=1000j``. We see that
    `yv` returns nan but `yve` returns a finite number:

    >>> import numpy as np
    >>> from scipy.special import yv, yve
    >>> v = 1
    >>> z = 1000j
    >>> yv(v, z), yve(v, z)
    ((nan+nanj), (-0.012610930256928629+7.721967686709076e-19j))

    For real arguments for `z`, `yve` returns the same as `yv` up to
    floating point errors.

    >>> v, z = 1, 1000
    >>> yv(v, z), yve(v, z)
    (-0.02478433129235178, -0.02478433129235179)

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

    >>> yve([1, 2, 3], 1j)
    array([-0.20791042+0.14096627j,  0.38053618-0.04993878j,
           0.00815531-1.66311097j])

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

    >>> yve(1, np.array([1j, 2j, 3j]))
    array([-0.20791042+0.14096627j, -0.21526929+0.01205044j,
           -0.19682671+0.00127278j])

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

    >>> v = np.array([[1], [2]])
    >>> z = np.array([3j, 4j, 5j])
    >>> v.shape, z.shape
    ((2, 1), (3,))

    >>> yve(v, z)
    array([[-1.96826713e-01+1.27277544e-03j, -1.78750840e-01+1.45558819e-04j,
            -1.63972267e-01+1.73494110e-05j],
           [1.94960056e-03-1.11782545e-01j,  2.02902325e-04-1.17626501e-01j,
            2.27727687e-05-1.17951906e-01j]])
    

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