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.

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()