# 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])


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