# 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.

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

References



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()
```