scipy.special.exp1

scipy.special.exp1(z, out=None) = <ufunc 'exp1'>

Exponential integral E1.

For complex \(z \ne 0\) the exponential integral can be defined as [1]

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

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 \(Ei\)

expn

generalization of \(E_1\)

Notes

For \(x > 0\) it is related to the exponential integral \(Ei\) (see expi) via the relation

\[E_1(x) = -Ei(-x).\]

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])