scipy.special.expi#

scipy.special.expi(x, out=None) = <ufunc 'expi'>#

Exponential integral Ei.

For real \(x\), the exponential integral is defined as [1]

\[Ei(x) = \int_{-\infty}^x \frac{e^t}{t} dt.\]

For \(x > 0\) the integral is understood as a Cauchy principal value.

It is extended to the complex plane by analytic continuation of the function on the interval \((0, \infty)\). The complex variant has a branch cut on the negative real axis.

Parameters:

xarray_like: Real or complex valued argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Values of the exponential integral

See also

exp1: Exponential integral \(E_1\)
expn: Generalized exponential integral \(E_n\)

Notes

The exponential integrals \(E_1\) and \(Ei\) satisfy the relation

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

for \(x > 0\).

expi has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. 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

See Support for the array API standard for more information.

References

[1]

Digital Library of Mathematical Functions, 6.2.5 https://dlmf.nist.gov/6.2#E5

Examples

>>> import numpy as np
>>> import scipy.special as sc

It is related to exp1.

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

The complex variant has a branch cut on the negative real axis.

>>> sc.expi(-1 + 1e-12j)
(-0.21938393439552062+3.1415926535894254j)
>>> sc.expi(-1 - 1e-12j)
(-0.21938393439552062-3.1415926535894254j)

As the complex variant approaches the branch cut, the real parts approach the value of the real variant.

>>> sc.expi(-1)
-0.21938393439552062

The SciPy implementation returns the real variant for complex values on the branch cut.

>>> sc.expi(complex(-1, 0.0))
(-0.21938393439552062-0j)
>>> sc.expi(complex(-1, -0.0))
(-0.21938393439552062-0j)