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.

outndarray, 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])