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.

See also

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.

Array API Standard Support

i0 has support for Python Array API Standard compatible backends in addition to NumPy. 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

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable.

See Support for the array API standard for more information.

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()