# scipy.special.i0#

scipy.special.i0(x, out=None) = <ufunc 'i0'>#

Modified Bessel function of order 0.

Defined as,

$I_0(x) = \sum_{k=0}^\infty \frac{(x^2/4)^k}{(k!)^2} = J_0(\imath x),$

where $$J_0$$ is the Bessel function of the first kind of order 0.

Parameters:
xarray_like

Argument (float)

outndarray, optional

Optional output array for the function values

Returns:
Iscalar or ndarray

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

iv

Modified Bessel function of any order

i0e

Exponentially scaled 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.

This function is a wrapper for the Cephes [1] routine i0.

References

[1]

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

Examples

Calculate the function at one point:

>>> from scipy.special import i0
>>> i0(1.)
1.2660658777520082


Calculate at several points:

>>> import numpy as np
>>> i0(np.array([-2., 0., 3.5]))
array([2.2795853 , 1.        , 7.37820343])


Plot the function from -10 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i0(x)
>>> ax.plot(x, y)
>>> plt.show()