# scipy.special.kvp

scipy.special.kvp(v, z, n=1)

  Compute derivatives of real-order modified Bessel function Kv(z)

  Kv(z) is the modified Bessel function of the second kind. Derivative is calculated with respect to z.

  Parameters:
    v array_like of float

      Order of Bessel function

    z array_like of complex

      Argument at which to evaluate the derivative

    n int, default 1

      Order of derivative. For 0 returns the Bessel function `kv` itself.

  Returns:
    out ndarray

      The results

  See also

  `kv`

  Notes

  The derivative is computed using the relation DLFM 10.29.5 [2].

  References

  [1]

  Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996, chapter 6. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

  [2]

  NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.29.E5

  Examples

  Compute the modified bessel function of the second kind of order 0 and its first two derivatives at 1.

      >>> from scipy.special import kvp
      >>> kvp(0, 1, 0), kvp(0, 1, 1), kvp(0, 1, 2)
      (0.42102443824070834, -0.6019072301972346, 1.0229316684379428)

  Compute the first derivative of the modified Bessel function of the second kind for several orders at 1 by providing an array for v.

      >>> kvp([0, 1, 2], 1, 1)
      array([-0.60190723, -1.02293167, -3.85158503])

  Compute the first derivative of the modified Bessel function of the second kind of order 0 at several points by providing an array for z.

      >>> import numpy as np
      >>> points = np.array([0.5, 1.5, 3.])
      >>> kvp(0, points, 1)
      array([-1.65644112, -0.2773878 , -0.04015643])

  Plot the modified bessel function of the second kind and its first three derivatives.

      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(0, 5, 1000)
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, kvp(1, x, 0), label=r"$K_1$")
      >>> ax.plot(x, kvp(1, x, 1), label=r"$K_1'$")
      >>> ax.plot(x, kvp(1, x, 2), label=r"$K_1''$")
      >>> ax.plot(x, kvp(1, x, 3), label=r"$K_1'''$")
      >>> ax.set_ylim(-2.5, 2.5)
      >>> plt.legend()
      >>> plt.show()