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:

zarray_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

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

Array API Standard Support

exp1 has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. exp1 does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Digital Library of Mathematical Functions, 6.2.1 https://dlmf.nist.gov/6.2#E1

Examples

>>> import numpy as np
>>> 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])