scipy.special.expn#

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

Generalized exponential integral En.

For integer \(n \geq 0\) and real \(x \geq 0\) the generalized exponential integral is defined as [dlmf]

\[E_n(x) = x^{n - 1} \int_x^\infty \frac{e^{-t}}{t^n} dt.\]

Parameters:

narray_like: Non-negative integers
xarray_like: Real argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: Values of the generalized exponential integral

See also

exp1: special case of \(E_n\) for \(n = 1\)
expi: related to \(E_n\) when \(n = 1\)

Notes

Array API Standard Support

expn 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

[dlmf]

Digital Library of Mathematical Functions, 8.19.2 https://dlmf.nist.gov/8.19#E2

Examples

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

Its domain is nonnegative n and x.

>>> sc.expn(-1, 1.0), sc.expn(1, -1.0)
(nan, nan)

It has a pole at x = 0 for n = 1, 2; for larger n it is equal to 1 / (n - 1).

>>> sc.expn([0, 1, 2, 3, 4], 0)
array([       inf,        inf, 1.        , 0.5       , 0.33333333])

For n equal to 0 it reduces to exp(-x) / x.

>>> x = np.array([1, 2, 3, 4])
>>> sc.expn(0, x)
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])
>>> np.exp(-x) / x
array([0.36787944, 0.06766764, 0.01659569, 0.00457891])

For n equal to 1 it reduces to exp1.

>>> sc.expn(1, x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])
>>> sc.exp1(x)
array([0.21938393, 0.04890051, 0.01304838, 0.00377935])