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Module « scipy.special »

Fonction struve - module scipy.special

Signature de la fonction struve

def struve(*args, **kwargs) 

Description

help(scipy.special.struve)

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


    struve(v, x, out=None)

    Struve function.

    Return the value of the Struve function of order `v` at `x`.  The Struve
    function is defined as,

    .. math::
        H_v(x) = (z/2)^{v + 1} \sum_{n=0}^\infty
        \frac{(-1)^n (z/2)^{2n}}{\Gamma(n + \frac{3}{2}) \Gamma(n + v + \frac{3}{2})},

    where :math:`\Gamma` is the gamma function.

    Parameters
    ----------
    v : array_like
        Order of the Struve function (float).
    x : array_like
        Argument of the Struve function (float; must be positive unless `v` is
        an integer).
    out : ndarray, optional
        Optional output array for the function results

    Returns
    -------
    H : scalar or ndarray
        Value of the Struve function of order `v` at `x`.

    See Also
    --------
    modstruve: Modified Struve function

    Notes
    -----
    Three methods discussed in [1]_ are used to evaluate the Struve function:

    - power series
    - expansion in Bessel functions (if :math:`|z| < |v| + 20`)
    - asymptotic large-z expansion (if :math:`z \geq 0.7v + 12`)

    Rounding errors are estimated based on the largest terms in the sums, and
    the result associated with the smallest error is returned.

    References
    ----------
    .. [1] NIST Digital Library of Mathematical Functions
           https://dlmf.nist.gov/11

    Examples
    --------
    Calculate the Struve function of order 1 at 2.

    >>> import numpy as np
    >>> from scipy.special import struve
    >>> import matplotlib.pyplot as plt
    >>> struve(1, 2.)
    0.6467637282835622

    Calculate the Struve function at 2 for orders 1, 2 and 3 by providing
    a list for the order parameter `v`.

    >>> struve([1, 2, 3], 2.)
    array([0.64676373, 0.28031806, 0.08363767])

    Calculate the Struve function of order 1 for several points by providing
    an array for `x`.

    >>> points = np.array([2., 5., 8.])
    >>> struve(1, points)
    array([0.64676373, 0.80781195, 0.48811605])

    Compute the Struve function for several orders at several points by
    providing arrays for `v` and `z`. The arrays have to be broadcastable
    to the correct shapes.

    >>> orders = np.array([[1], [2], [3]])
    >>> points.shape, orders.shape
    ((3,), (3, 1))

    >>> struve(orders, points)
    array([[0.64676373, 0.80781195, 0.48811605],
           [0.28031806, 1.56937455, 1.51769363],
           [0.08363767, 1.50872065, 2.98697513]])

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

    >>> fig, ax = plt.subplots()
    >>> x = np.linspace(-10., 10., 1000)
    >>> for i in range(4):
    ...     ax.plot(x, struve(i, x), label=f'$H_{i!r}$')
    >>> ax.legend(ncol=2)
    >>> ax.set_xlim(-10, 10)
    >>> ax.set_title(r"Struve functions $H_{\nu}$")
    >>> plt.show()
    

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