scipy.special.i0e(x, out=None) = <ufunc 'i0e'>#

Exponentially scaled modified Bessel function of order 0.

Defined as:

i0e(x) = exp(-abs(x)) * i0(x).

Argument (float)

outndarray, optional

Optional output array for the function values

Iscalar or ndarray

Value of the exponentially scaled modified Bessel function of order 0 at x.

See also


Modified Bessel function of the first kind


Modified Bessel function of order 0


The range is partitioned into the two intervals [0, 8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. The polynomial expansions used are the same as those in i0, but they are not multiplied by the dominant exponential factor.

This function is a wrapper for the Cephes [1] routine i0e.



Cephes Mathematical Functions Library,


Calculate the function at one point:

>>> from scipy.special import i0e
>>> i0e(1.)

Calculate the function at several points:

>>> import numpy as np
>>> i0e(np.array([-2., 0., 3.]))
array([0.30850832, 1.        , 0.24300035])

Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0e(x)
>>> ax.plot(x, y)

Exponentially scaled Bessel functions are useful for large arguments for which the unscaled Bessel functions overflow or lose precision. In the following example i0 returns infinity whereas i0e still returns a finite number.

>>> from scipy.special import i0
>>> i0(1000.), i0e(1000.)
(inf, 0.012617240455891257)