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
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 experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variableSCIPY_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
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()