scipy.special.expn — SciPy v1.15.2 Manual (original) (raw)

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 \(n = 1\)

References

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])