Skip to main content
`Ctrl`+`K`
scipy.special.

# yvp

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

  Compute derivatives of Bessel functions of the second kind.

  Compute the nth derivative of the Bessel function Yv with respect to z.

  Parameters:
    v array_like of float

      Order of Bessel function

    z complex

      Argument at which to evaluate the derivative

    n int, default 1

      Order of derivative. For 0 returns the BEssel function `yv`

  Returns:
    scalar or ndarray

      nth derivative of the Bessel function.

  See also

  `yv`

    Bessel functions of the second kind

  Notes

  The derivative is computed using the relation DLFM 10.6.7 [2].

  References

  [1]

  Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996, chapter 5. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

  [2]

  NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.6.E7

  Examples

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

      >>> from scipy.special import yvp
      >>> yvp(0, 1, 0), yvp(0, 1, 1), yvp(0, 1, 2)
      (0.088256964215677, 0.7812128213002889, -0.8694697855159659)

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

      >>> yvp([0, 1, 2], 1, 1)
      array([0.78121282, 0.86946979, 2.52015239])

  Compute the first derivative of the 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.])
      >>> yvp(0, points, 1)
      array([ 1.47147239,  0.41230863, -0.32467442])

  Plot the Bessel function of the second kind of order 1 and its first three derivatives.

      >>> import matplotlib.pyplot as plt
      >>> x = np.linspace(0, 5, 1000)
      >>> x[0] += 1e-15
      >>> fig, ax = plt.subplots()
      >>> ax.plot(x, yvp(1, x, 0), label=r"$Y_1$")
      >>> ax.plot(x, yvp(1, x, 1), label=r"$Y_1'$")
      >>> ax.plot(x, yvp(1, x, 2), label=r"$Y_1''$")
      >>> ax.plot(x, yvp(1, x, 3), label=r"$Y_1'''$")
      >>> ax.set_ylim(-10, 10)
      >>> plt.legend()
      >>> plt.show()