Skip to main content
`Ctrl`+`K`

# scipy.special.k1e

scipy.special.k1e(x, out=None) = <ufunc 'k1e'>

  Exponentially scaled modified Bessel function K of order 1

  Defined as:

      k1e(x) = exp(x) * k1(x)

  Parameters:
    xarray_like

      Argument (float)

    outndarray, optional

      Optional output array for the function values

  Returns:
    Kscalar or ndarray

      Value of the exponentially scaled modified Bessel function K of order 1 at x.

  See also

  `kv`

    Modified Bessel function of the second kind of any order

  `k1`

    Modified Bessel function of the second kind 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 `k1e`.

  References

  [1]

  Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

  Examples

  In the following example `k1` returns 0 whereas `k1e` still returns a useful floating point number.

      >>> from scipy.special import k1, k1e
      >>> k1(1000.), k1e(1000.)
      (0., 0.03964813081296021)

  Calculate the function at several points by providing a NumPy array or list for x:

      >>> import numpy as np
      >>> k1e(np.array([0.5, 2., 3.]))
      array([2.73100971, 1.03347685, 0.80656348])

  Plot the function from 0 to 10.

      >>> import matplotlib.pyplot as plt
      >>> fig, ax = plt.subplots()
      >>> x = np.linspace(0., 10., 1000)
      >>> y = k1e(x)
      >>> ax.plot(x, y)
      >>> plt.show()