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 \(n = 1\)

References

dlmf

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

Examples

>>> 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])