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Module « scipy.special »

Fonction expi - module scipy.special

Signature de la fonction expi

def expi(*args, **kwargs) 

Description

help(scipy.special.expi)

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


    expi(x, out=None)

    Exponential integral Ei.

    For real :math:`x`, the exponential integral is defined as [1]_

    .. math::

        Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt.

    For :math:`x > 0` the integral is understood as a Cauchy principal
    value.

    It is extended to the complex plane by analytic continuation of
    the function on the interval :math:`(0, \infty)`. The complex
    variant has a branch cut on the negative real axis.

    Parameters
    ----------
    x : array_like
        Real or complex valued argument
    out : ndarray, optional
        Optional output array for the function results

    Returns
    -------
    scalar or ndarray
        Values of the exponential integral

    See Also
    --------
    exp1 : Exponential integral :math:`E_1`
    expn : Generalized exponential integral :math:`E_n`

    Notes
    -----
    The exponential integrals :math:`E_1` and :math:`Ei` satisfy the
    relation

    .. math::

        E_1(x) = -Ei(-x)

    for :math:`x > 0`.

    References
    ----------
    .. [1] Digital Library of Mathematical Functions, 6.2.5
           https://dlmf.nist.gov/6.2#E5

    Examples
    --------
    >>> import numpy as np
    >>> import scipy.special as sc

    It is related to `exp1`.

    >>> x = np.array([1, 2, 3, 4])
    >>> -sc.expi(-x)
    array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
    >>> sc.exp1(x)
    array([0.21938393, 0.04890051, 0.01304838, 0.00377935])

    The complex variant has a branch cut on the negative real axis.

    >>> sc.expi(-1 + 1e-12j)
    (-0.21938393439552062+3.1415926535894254j)
    >>> sc.expi(-1 - 1e-12j)
    (-0.21938393439552062-3.1415926535894254j)

    As the complex variant approaches the branch cut, the real parts
    approach the value of the real variant.

    >>> sc.expi(-1)
    -0.21938393439552062

    The SciPy implementation returns the real variant for complex
    values on the branch cut.

    >>> sc.expi(complex(-1, 0.0))
    (-0.21938393439552062-0j)
    >>> sc.expi(complex(-1, -0.0))
    (-0.21938393439552062-0j)

    

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