Module « scipy.special »

Fonction expi - module scipy.special

Signature de la fonction expi

Description

expi.__doc__

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

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 principle
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

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

.. math::

    E_1(x) = -Ei(-x)

for :math:`x > 0`.

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

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

Examples
--------
>>> 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.

>>> import scipy.special as sc
>>> 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)