Module « scipy.special »

Fonction exp1 - module scipy.special

Signature de la fonction exp1

Description

exp1.__doc__

exp1(x, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

exp1(z, out=None)

Exponential integral E1.

For complex :math:`z \ne 0` the exponential integral can be defined as
[1]_

.. math::

   E_1(z) = \int_z^\infty \frac{e^{-t}}{t} dt,

where the path of the integral does not cross the negative real
axis or pass through the origin.

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

Returns
-------
scalar or ndarray
    Values of the exponential integral E1

See Also
--------
expi : exponential integral :math:`Ei`
expn : generalization of :math:`E_1`

Notes
-----
For :math:`x > 0` it is related to the exponential integral
:math:`Ei` (see `expi`) via the relation

.. math::

   E_1(x) = -Ei(-x).

References
----------
.. [1] Digital Library of Mathematical Functions, 6.2.1
       https://dlmf.nist.gov/6.2#E1

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

It has a pole at 0.

>>> sc.exp1(0)
inf

It has a branch cut on the negative real axis.

>>> sc.exp1(-1)
nan
>>> sc.exp1(complex(-1, 0))
(-1.8951178163559368-3.141592653589793j)
>>> sc.exp1(complex(-1, -0.0))
(-1.8951178163559368+3.141592653589793j)

It approaches 0 along the positive real axis.

>>> sc.exp1([1, 10, 100, 1000])
array([2.19383934e-01, 4.15696893e-06, 3.68359776e-46, 0.00000000e+00])

It is related to `expi`.

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