scipy.special.i1#

scipy.special.i1(x, out=None) = <ufunc 'i1'>#

Modified Bessel function of order 1.

Defined as,

\[I_1(x) = \frac{1}{2}x \sum_{k=0}^\infty \frac{(x^2/4)^k}{k! (k + 1)!} = -\imath J_1(\imath x),\]

where \(J_1\) is the Bessel function of the first kind of order 1.

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 1 at x.

See also

iv: Modified Bessel function of the first kind
i1e: Exponentially scaled modified Bessel function of order 1

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 i1.

Array API Standard Support

i1 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

Calculate the function at one point:

>>> from scipy.special import i1
>>> i1(1.)
0.5651591039924851

Calculate the function at several points:

>>> import numpy as np
>>> i1(np.array([-2., 0., 6.]))
array([-1.59063685,  0.        , 61.34193678])

Plot the function between -10 and 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(-10., 10., 1000)
>>> y = i1(x)
>>> ax.plot(x, y)
>>> plt.show()