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Module « scipy.special »

Fonction iv - module scipy.special

Signature de la fonction iv

def iv(*args, **kwargs) 

Description

help(scipy.special.iv)

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


    iv(v, z, out=None)

    Modified Bessel function of the first kind of real order.

    Parameters
    ----------
    v : array_like
        Order. If `z` is of real type and negative, `v` must be integer
        valued.
    z : array_like of float or complex
        Argument.
    out : ndarray, optional
        Optional output array for the function values

    Returns
    -------
    scalar or ndarray
        Values of the modified Bessel function.

    See Also
    --------
    ive : This function with leading exponential behavior stripped off.
    i0 : Faster version of this function for order 0.
    i1 : Faster version of this function for order 1.

    Notes
    -----
    For real `z` and :math:`v \in [-50, 50]`, the evaluation is carried out
    using Temme's method [1]_.  For larger orders, uniform asymptotic
    expansions are applied.

    For complex `z` and positive `v`, the AMOS [2]_ `zbesi` routine is
    called. It uses a power series for small `z`, the asymptotic expansion
    for large `abs(z)`, the Miller algorithm normalized by the Wronskian
    and a Neumann series for intermediate magnitudes, and the uniform
    asymptotic expansions for :math:`I_v(z)` and :math:`J_v(z)` for large
    orders. Backward recurrence is used to generate sequences or reduce
    orders when necessary.

    The calculations above are done in the right half plane and continued
    into the left half plane by the formula,

    .. math:: I_v(z \exp(\pm\imath\pi)) = \exp(\pm\pi v) I_v(z)

    (valid when the real part of `z` is positive).  For negative `v`, the
    formula

    .. math:: I_{-v}(z) = I_v(z) + \frac{2}{\pi} \sin(\pi v) K_v(z)

    is used, where :math:`K_v(z)` is the modified Bessel function of the
    second kind, evaluated using the AMOS routine `zbesk`.

    References
    ----------
    .. [1] Temme, Journal of Computational Physics, vol 21, 343 (1976)
    .. [2] 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 iv
    >>> iv(0, 1.)
    1.2660658777520084

    Evaluate the function at one point for different orders.

    >>> iv(0, 1.), iv(1, 1.), iv(1.5, 1.)
    (1.2660658777520084, 0.565159103992485, 0.2935253263474798)

    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:

    >>> iv([0, 1, 1.5], 1.)
    array([1.26606588, 0.5651591 , 0.29352533])

    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.])
    >>> iv(0, points)
    array([2.2795853 , 1.        , 4.88079259])

    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)

    >>> iv(orders, points)
    array([[ 2.2795853 ,  1.        ,  4.88079259],
           [-1.59063685,  0.        ,  3.95337022]])

    Plot the functions of order 0 to 3 from -5 to 5.

    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots()
    >>> x = np.linspace(-5., 5., 1000)
    >>> for i in range(4):
    ...     ax.plot(x, iv(i, x), label=f'$I_{i!r}$')
    >>> ax.legend()
    >>> plt.show()
    

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