scipy.special.k1#

scipy.special.k1(x, out=None) = <ufunc 'k1'>#

Modified Bessel function of the second kind of order 1, \(K_1(x)\).

Parameters:

xarray_like: Argument (float)
outndarray, optional: Optional output array for the function values

Returns:

Kscalar or ndarray: Value of the modified Bessel function K of order 1 at x.

See also

kv: Modified Bessel function of the second kind of any order
k1e: Exponentially scaled modified Bessel function K of order 1

Notes

The range is partitioned into the two intervals [0, 2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval.

This function is a wrapper for the Cephes [1] routine k1.

Array API Standard Support

k1 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	⚠️ no JIT	⛔
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 k1
>>> k1(1.)
0.6019072301972346

Calculate the function at several points:

>>> import numpy as np
>>> k1(np.array([0.5, 2., 3.]))
array([1.65644112, 0.13986588, 0.04015643])

Plot the function from 0 to 10.

>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots()
>>> x = np.linspace(0., 10., 1000)
>>> y = k1(x)
>>> ax.plot(x, y)
>>> plt.show()