scipy.special.i0e#

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).

Parameters:

xarray_like: Argument (float)
outndarray, optional: Optional output array for the function values

Returns:

Iscalar or ndarray: Value of the exponentially scaled modified Bessel function of order 0 at x.

See also

iv: Modified Bessel function of the first kind
i0: Modified Bessel function of order 0

Notes

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. i0e is useful for large arguments x: for these, i0 quickly overflows.

i0e 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]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

In the following example i0 returns infinity whereas i0e still returns a finite number.

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

Calculate the function at several points by providing a NumPy array or list for x:

>>> 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)
>>> plt.show()