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# 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`.

  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()